MAT 170 - Precalculus - Algebraic Foundations
1. Algebraic Foundations
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
• Factoring Techniques
o Difference of Squares: For expressions like a2−b2a^2 - b^2a2−b2, factor into
(a+b)(a−b)(a + b)(a - b)(a+b)(a−b).
▪ Example: x2−16=(x+4)(x−4)x^2 - 16 = (x + 4)(x - 4)x2−16=(x+4)(x−4).
o Trinomials: For expressions of the form ax2+bx+cax^2 + bx + cax2+bx+c, use the
factoring method or quadratic formula if necessary.
▪ Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3).
o Grouping: Useful for polynomials with four terms.
▪ Example: For x3+x2+3x+3x^3 + x^2 + 3x + 3x3+x2+3x+3, factor out common
terms: x2(x+1)+3(x+1)=(x+1)(x2+3)x^2(x + 1) + 3(x + 1) = (x + 1)(x^2 +
3)x2(x+1)+3(x+1)=(x+1)(x2+3).
• Quadratic Formula and Completing the Square
o Quadratic Formula: Solve ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 with
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac
▪ Example: For 2x2−4x−6=02x^2 - 4x - 6 = 02x2−4x−6=0, substitute a=2a = 2a=2,
b=−4b = -4b=−4, c=−6c = -6c=−6 into the formula.
o Completing the Square: A method to rewrite quadratics in vertex form, helpful in
graphing and solving equations.
▪ Example: For x2+6x+5=0x^2 + 6x + 5 = 0x2+6x+5=0, rewrite as (x+3)2−4=0(x +
3)^2 - 4 = 0(x+3)2−4=0.
• Exponents and Radicals
o Review properties of exponents and practice simplifications, such as:
am
⋅
an=am+n,am/an=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad a^m / a^n
= a^{m-n}, \quad (a^m)^n = a^{mn}am
⋅
an=am+n,am/an=am−n,(am)n=amn
o Radicals: Converting between radicals and fractional exponents (e.g.,
x3=x1/3\sqrt[3]{x} = x^{1/3}3x=x1/3) and simplifying expressions involving square
roots.
2. Functions and Their Properties
• Types of Functions:
o Linear, quadratic, polynomial, rational, exponential, and logarithmic functions.
o Examples: Graphs and properties of each, like the general form of linear functions
y=mx+by = mx + by=mx+b and quadratic functions y=ax2+bx+cy = ax^2 + bx +
cy=ax2+bx+c.
• Domain and Range:
o The domain is the set of all possible input values for which the function is defined.
o Example: For f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2, the domain is x≥2x \geq 2x≥2 because
square roots of negative numbers are undefined.
• Composite Functions:
(f
∘
g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f
∘
g)(x)=f(g(x))
o Example: For f(x)=x+2f(x) = x + 2f(x)=x+2 and g(x)=3xg(x) = 3xg(x)=3x, find (f
∘
g)(x)(f
\circ g)(x)(f
∘
g)(x) and (g
∘
f)(x)(g \circ f)(x)(g
∘
f)(x).
• Inverse Functions:
o Finding inverses by swapping xxx and yyy and solving for yyy.
o Verify with f(f−1(x))=xf(f^{-1}(x)) = xf(f−1(x))=x.
o Example: If f(x)=2x+3f(x) = 2x + 3f(x)=2x+3, the inverse f−1(x)=x−32f^{-1}(x) = \frac{x
- 3}{2}f−1(x)=2x−3.
3. Polynomial and Rational Functions
• Factoring and Roots:
o Factor Theorem: If f(a)=0f(a) = 0f(a)=0, then x−ax - ax−a is a factor of f(x)f(x)f(x).
o Synthetic Division: A simplified division method to factor polynomials.
o Example: Use synthetic division to determine if x−2x - 2x−2 is a factor of
x3−4x2+5x−2x^3 - 4x^2 + 5x - 2x3−4x2+5x−2.
• Rational Function Asymptotes:
o Vertical Asymptotes: Where the function becomes undefined (typically where the
denominator is zero).
▪ Example: For f(x)=2x−3f(x) = \frac{2}{x - 3}f(x)=x−32, there is a vertical
asymptote at x=3x = 3x=3.
o Horizontal Asymptotes: Based on the degrees of numerator and denominator.
▪ If the degree of the numerator is less than the denominator, y=0y = 0y=0 is the
asymptote.
o End Behavior: Leading term test for polynomial functions.
4. Exponential and Logarithmic Functions
• Laws of Exponents and Properties of Logarithms:
o For simplification and solving, use properties like:
logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb
(y)
o Natural Logarithm (ln): Useful in continuous growth problems.
o Solving Exponential and Logarithmic Equations:
▪ Example: For 2x=82^x = 82x=8, rewrite as 2x=232^x = 2^32x=23 so x=3x =
3x=3.
▪ For logarithmic equations, use properties to isolate the variable.
5. Trigonometry
• Basic Identities: Key identities to simplify trigonometric expressions.
o sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
o tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
• Unit Circle:
o Memorize angles in radians and their coordinates.
o Examples: 000, π6\frac{\pi}{6}6π, π4\frac{\pi}{4}4π, π3\frac{\pi}{3}3π,
π2\frac{\pi}{2}2π and their corresponding (x,y)(x, y)(x,y) values.
6. Analytic Geometry
• Circles and Conic Sections:
o Equation of a circle: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2,
with (h,k)(h, k)(h,k) as the center and rrr as the radius.
o Ellipses and Hyperbolas: Standard equations and key features.
o Example: For a circle with equation (x−3)2+(y+2)2=9(x - 3)^2 + (y + 2)^2 =
9(x−3)2+(y+2)2=9, the center is (3,−2)(3, -2)(3,−2) and radius r=3r = 3r=3.
• Distance and Midpoint Formulas:
o Distance Formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2
−x1)2+(y2−y1)2
o Midpoint Formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 +
y_2}{2} \right)M=(2x1+x2,2y1+y2)
7. Sequences and Series
• Arithmetic Sequence: Formula an=a1+(n−1)da_n = a_1 + (n - 1)dan=a1+(n−1)d.
o Example: Find the 5th term when a1=2a_1 = 2a1=2 and d=3d = 3d=3.
• Geometric Sequence: Formula an=a1
⋅
rn−1a_n = a_1 \cdot r^{n - 1}an=a1
⋅
rn−1.
• Sum of Sequences:
o Arithmetic Sum: Sn=n2(a1+an)S_n = \frac{n}{2} (a_1 + a_n)Sn=2n(a1+an).
o Geometric Sum: Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r}Sn=a11−r1−rn, r≠1r \neq
1r =1.
8. Introduction to Limits
• Evaluating Limits:
o Use substitution where possible; for indeterminate forms, factor or use L'Hôpital's Rule
if introduced.
• One-Sided Limits:
o Limits from the left (x→a−x \to a^-x→a−) and right (x→a+x \to a^+x→a+).
Problem-Solving Tips and Strategies
• Graph Analysis: Practice graphing different types of functions to understand behavior.
• Equation Solving Techniques: Familiarize with multiple approaches, such as factoring, using
identities, and properties.
sv
• Checking Work: Verifying solutions, especially for identities and equation solutions, by
substitution.
