please answer a minimum of 3 OF 6 question pairs. you must answer one question out of each pair with a minimum of one paragraph.
dream86
Ch2lecture.pptxCopyright © Cengage Learning. All rights reserved.
‹#›
1
Copyright © Cengage Learning. All rights reserved.
Two-Dimensional Coordinate System and Graphs
SECTION 2.1
‹#›
2
Cartesian Coordinate Systems
‹#›
3
Cartesian Coordinate Systems
Each point on a coordinate axis is associated with a number called its coordinate.
Each point on a flat, two-dimensional surface, called a coordinate plane or xy-plane, is associated with an ordered pair of numbers called coordinates of the point.
Ordered pairs are denoted by (a, b), where the real number a is the x-coordinate or abscissa and the real number b is the y-coordinate or ordinate.
‹#›
4
Cartesian Coordinate Systems
The coordinates of a point are determined by the point’s position relative to a horizontal coordinate axis called the x-axis and a vertical coordinate axis called the y-axis. The axes intersect at the point (0, 0), called the origin.
In Figure 2.1, the axes are labeled such that positive numbers appear to the right of the origin on the x-axis and above the origin on the y-axis.
Figure 2.1
‹#›
5
Cartesian Coordinate Systems
The four regions formed by the axes are called quadrants and are numbered counterclockwise. This two-dimensional coordinate system is referred to as a Cartesian coordinate system in honor of René Descartes.
To plot a point P(a, b) means to draw a dot at its location in the coordinate plane.
In Figure 2.2, we have plotted the points (4, 3), (–3, 1), (–2, –3), (3, –2), (0, 1), (1, 3), and (3,1).
Figure 2.2
‹#›
6
Cartesian Coordinate Systems
The order in which the coordinates of an ordered pair are listed is important. Figure 2.2 shows that (1, 3) and (3, 1) do not denote the same point.
Data often are displayed in visual form as a set of points called a scatter plot.
‹#›
7
Cartesian Coordinate Systems
For instance, the scatter plot in Figure 2.3 shows the growth in text messaging during the years 2005 to 2011, with each point representing the data from a 12-month period ending in June.
Figure 2.3
Source: CTIA.
‹#›
8
Cartesian Coordinate Systems
In some instances, it is important to know when two ordered pairs are equal.
Definition of the Equality of Ordered Pairs
The ordered pairs (a, b) and (c, d) are equal if and only if a = c and b = d.
Example
If (3, y) = (x, –2), then x = 3 and y = –2.
‹#›
9
Distance and Midpoint Formulas
‹#›
10
Distance and Midpoint Formulas
The Cartesian coordinate system makes it possible to combine the concepts of algebra and geometry into a branch of mathematics called analytic geometry.
The distance between two points on a horizontal line is the absolute value of the difference between the x-coordinates of the two points.
The distance between two points on a vertical line is the absolute value of the difference between the y-coordinates of the two points.
‹#›
11
Distance and Midpoint Formulas
For example, as shown in Figure 2.4, the distance d between the points with coordinates (1, 2) and (1, –3) is d = | 2 – (–3) | = 5.
Figure 2.4
‹#›
12
Distance and Midpoint Formulas
If two points are not on a horizontal or vertical line, then a distance formula for the distance between the two points can be developed as follows.
The distance between the points P1(x1, y1) and P2(x2, y2) in Figure 2.5 is the length of the hypotenuse of a right triangle whose sides are horizontal and vertical line segments that measure | x2 – x1 | and | y2 – y1 |, respectively.
Figure 2.5
‹#›
13
Distance and Midpoint Formulas
Applying the Pythagorean Theorem to this triangle produces
Use the square root procedure. Because d is nonnegative, the negative root is not listed.
| x2 – x1 |2 = (x2 – x1)2 and
| y2 – y1 |2 = (y2 – y1)2
‹#›
14
Distance and Midpoint Formulas
Thus we have established the following theorem.
Distance Formula
The distance d(P1, P2) between the points P1(x1, y1) and P2(x2, y2) is
‹#›
15
Distance and Midpoint Formulas
Example
The distance between P1(–3, 4) and P2(7, 2) is given by
‹#›
16
Distance and Midpoint Formulas
The midpoint M of a line segment is the point on the line segment that is equidistant from the endpoints P1(x1, y1) and P2(x2, y2) of the segment. See Figure 2.6.
Midpoint Formula
The midpoint M of the line segment from P1(x1, y1) to P2(x2, y2) is given by
Figure 2.6
‹#›
17
Distance and Midpoint Formulas
Example
The midpoint of the line segment between P1(–2, 6) and P2(3, 4) is given by
‹#›
18
Distance and Midpoint Formulas
The midpoint formula states that the x-coordinate of the midpoint of a line segment is the average of the x-coordinates of the endpoints of the line segment and that the y-coordinate of the midpoint of a line segment is the average of the y-coordinates of the endpoints of the line segment.
‹#›
19
Example 1 – Find the Midpoint and Length of a Line Segment
Find the midpoint and the length of the line segment connecting the points whose coordinates are P1(– 4, 3) and P2(4, –2).
Solution:
‹#›
20
Example 1 – Solution
cont’d
‹#›
21
Graph of an Equation
‹#›
22
Graph of an Equation
The equations below are equations in two variables.
y = 3x3 – 4x + 2 x2 + y2 = 25
The solution of an equation in two variables is an ordered pair (x, y) whose coordinates satisfy the equation.
For instance, the ordered pairs (3, 4), (4, –3), and (0, 5) are some of the solutions of x2 + y2 = 25.
‹#›
23
Graph of an Equation
Generally, there are an infinite number of solutions of an equation in two variables. These solutions can be displayed in a graph.
Definition of the Graph of an Equation
The graph of an equation in the two variables x and y is the set of all points (x, y) whose coordinates satisfy the equation.
‹#›
24
Graph of an Equation
Consider y = 2x – 1. Substituting various values of x into the equation and solving for y produces some of the ordered pairs that satisfy the equation.
It is convenient to record the results in a table similar to the one shown below.
‹#›
25
Graph of an Equation
The graph of the ordered pairs is shown in Figure 2.7.
Figure 2.7
‹#›
26
Graph of an Equation
Choosing some noninteger values of x produces more
ordered pairs to graph, such as and ,
as shown in Figure 2.8.
Figure 2.8
‹#›
27
Graph of an Equation
Using still other values of x would add even more ordered pairs to graph. The result would be so many dots that the graph would appear as the straight line shown in Figure 2.9, which is the graph of y = 2x – 1.
Figure 2.9
‹#›
28
Example 2 – Draw a Graph by Plotting Points
Graph: –x2 + y = 1
Solution:
Solve the equation for y.
y = x2 + 1
Select values of x and use the equation to calculate y. Choose enough values of x so that an accurate graph can be drawn.
‹#›
29
Example 2 – Solution
Plot the points and draw a curve through them. See Figure 2.10.
cont’d
Figure 2.10
‹#›
30
Intercepts
‹#›
31
Intercepts
On a graph, any point that has an x- or a y-coordinate of zero is called an intercept of the graph, because it is at this point that the graph intersects the x- or the y-axis.
Definitions of x-Intercepts and y-Intercepts
If (x1, 0) satisfies an equation in two variables, then the point whose coordinates are (x1, 0) is called an x-intercept of the graph of the equation.
If (0, y1) satisfies an equation in two variables, then the point whose coordinates are (0, y1) is called a y-intercept of the graph of the equation.
‹#›
32
Intercepts
To find the x-intercepts of the graph of an equation, let y = 0 and solve the equation for x.
To find the y-intercepts of the graph of an equation, let x = 0 and solve the equation for y.
‹#›
33
Example 5 – Find x- and y-Intercepts
Find the x- and y-intercepts of the graph of y = x2 – 2x – 3.
Algebraic Solution:
To find the y-intercept, let x = 0 and solve for y.
y = 02 – 2(0) – 3
= –3
To find the x-intercept, let y = 0 and solve for x.
0 = x2 – 2x – 3
0 = (x – 3)(x + 1)
‹#›
34
Example 5 – Solution
(x – 3) = 0 or (x + 1) = 0
x = 3 or x = –1
Because y = –3 when x = 0, (0,–3) is a y-intercept.
Because x = 3 or –1 when y = 0, (3, 0) and (–1, 0) are x-intercepts.
cont’d
‹#›
35
Example 5 – Solution
Figure 2.15 confirms that these three points are intercepts.
cont’d
Figure 2.15
‹#›
36
Circles, Their Equations, and Their Graphs
‹#›
37
Circles, Their Equations, and Their Graphs
Frequently you will sketch graphs by plotting points. However, some graphs can be sketched merely by recognizing the form of the equation.
A circle is an example of a curve whose graph you can sketch after you have inspected its equation.
Definition of a Circle
A circle is the set of points in a plane that are a fixed distance from a specified point. The fixed distance is the radius of the circle, and the specified point is the center of the circle.
‹#›
38
Circles, Their Equations, and Their Graphs
The standard form of the equation of a circle is derived by using the definition of a circle. To derive the standard form, we use the distance formula.
Figure 2.16 is a circle with center (h, k) and radius r.
Figure 2.16
‹#›
39
Circles, Their Equations, and Their Graphs
The point (x, y) is on the circle if and only if it is a distance of r units from the center (h, k). Thus (x, y) is on the circle if and only if
Standard Form of the Equation of a Circle
Let C(h, k) be the coordinates of the center of a circle of radius r. Then the standard form of the equation of a circle is given by
(x – h)2 + (y – k)2 = r 2
Square each side.
‹#›
40
Circles, Their Equations, and Their Graphs
Examples
The equation of the circle in standard form with center C(2, –5) and radius 4 is
(x – h)2 + (y – k)2 = r2
(x – 2)2 + (y – (–5))2 = 42
(x – 2)2 + (y + 5)2 = 16
Although (x – 2)2 + ( y – (–5))2 = 42 is actually standard form, we usually write the equation in simplest form as (x – 2)2 + ( y + 5)2 = 16.
h = 2, k = –5, r = 4
Simplify.
‹#›
41
Circles, Their Equations, and Their Graphs
The coordinates of the center and the radius of the circle whose equation is (x + 3)2 + (y – 4)2 = 7 are found by writing the standard form of the equation of the circle.
(x + 3)2 + (y – 4)2 = 7
(x – (–3))2 + (y – 4)2 =
The center is C(–3, 4) and the radius is r = .
Note that (x – (–3))2 = (x + 3)2;
= 7
‹#›
42
Circles, Their Equations, and Their Graphs
If a circle is centered at the origin (0, 0), then h = 0 and k = 0 and the standard form of the equation of the circle simplifies to
x2 + y2 = r 2
For instance, x2 + y2 = 9 is the equation of the circle with center at the origin and radius of .
Equation of a circle with center at
the origin and radius r
‹#›
43
Example 6 – Find the Standard Form of the Equation of a Circle
Find the standard form of the equation of a circle that has a diameter with endpoints whose coordinates are (–1, 4) and ( 5, –2), as shown in Figure 2.17.
Figure 2.17
‹#›
44
Example 6 – Solution
Find the midpoint of the diameter. The midpoint gives the coordinates for the center of the circle.
Coordinates of center:
Find the radius of the circle by finding the length of the line segment from the center to one of the endpoints of the diameter.
‹#›
45
Example 6 – Solution
Write the equation of the circle.
(x – h)2 + (y – k)2 = r2
(x – 2)2 + (y – 1)2 = 18
r2 = = 18
cont’d
‹#›
46
Circles, Their Equations, and Their Graphs
If we rewrite (x + 4)2 + (y + 2)2 = 25 by squaring and combining like terms, we produce
x2 + 8x + 16 + y2 + 4y + 4 = 25
x2 + y2 + 8x + 4y – 5 = 0
This form of the equation is known as the general form of the equation of a circle.
‹#›
47
Circles, Their Equations, and Their Graphs
By completing the square, it is always possible to rewrite an equation in the general form x2 + y2 + Ax + By + C = 0 in the standard form
(x – h)2 + (y – k)2 = s
for some number s.
If s > 0, the graph is a circle with radius .
If s = 0, the graph is the point (h, k).
If s < 0, the equation has no real solutions and there is no graph.
‹#›
48
Copyright © Cengage Learning. All rights reserved.
Introduction to Functions
SECTION 2.2
‹#›
49
Relations
‹#›
50
Relations
In many situations in science, business, and mathematics,
a correspondence exists between the two sets.
The correspondence is often defined by a table, an equation, or a graph, each of which can be viewed from a mathematical perspective as a set of ordered pairs. In mathematics, any set of ordered pairs is called a relation.
‹#›
51
Relations
Table 2.1 defines a correspondence between a set of percent scores and a set of letter grades.
For each score from 0 to 100, there corresponds only one letter grade.
Table 2.1
‹#›
52
Relations
The score 94% corresponds to the letter grade of A. Using ordered-pair notation, we record this correspondence as (94, A).
The equation d = 16t 2 indicates that the distance d that a rock falls (neglecting air resistance) corresponds to the time t that it has been falling.
For each nonnegative value t, the equation assigns only one value for the distance d.
‹#›
53
Relations
According to the equation, in 3 seconds a rock will fall 144 feet, which we record as (3, 144).
Some of the other ordered pairs determined by d = 16t 2
are (0, 0), (1, 16), (2, 64), and (2.5, 100).
Equation: d = 16t 2
If t = 3, then d = 16(3)2 = 144
‹#›
54
Relations
The graph in Figure 2.19, defines a correspondence between the length of a pendulum and the time it takes the pendulum to complete one oscillation.
Figure 2.19
‹#›
55
Relations
For each nonnegative pendulum length, the graph yields only one time.
According to the graph, a pendulum length of 2 feet yields an oscillation time of 1.6 seconds and a pendulum length of 4 feet yields an oscillation time of 2.2 seconds, where the time is measured to the nearest tenth of a second.
These results can be recorded as the ordered pairs (2, 1.6)
and (4, 2.2).
‹#›
56
Functions
‹#›
57
Functions
The table 2.1, earlier equation and the following graph each determine a special type of relation called a function.
Table 2.1
Figure 2.19
‹#›
58
Functions
Definition of a Function
A function is a set of ordered pairs in which no two ordered pairs have the same first coordinate and different second coordinates.
Although every function is a relation, not every relation is a function.
For instance, consider (94, A) from the grading correspondence. The first coordinate, 94, is paired with a second coordinate, A.
‹#›
59
Functions
It would not make sense to have 94 paired with A, (94, A), and 94 paired with B, (94, B).
The same first coordinate would be paired with two different second coordinates.
This would mean that two students with the same score received different grades, one student an A and the other a B!
‹#›
60
Functions
Functions may have ordered pairs with the same second coordinate.
For instance, (94, A) and (95, A) are both ordered pairs that belong to the function defined by Table 2.1.
Thus a function may have different first coordinates and the same second coordinate.
Table 2.1
‹#›
61
Functions
The equation d = 16t 2 represents a function because for each value of t there is only one value of d.
However, not every equation represents a function. For instance, y2 = 25 – x2 does not represent a function.
The ordered pairs (–3, 4) and (–3, –4) are both solutions of the equation.
However, these ordered pairs do not satisfy the definition of a function: There are two ordered pairs with the same first coordinate but different second coordinates.
‹#›
62
Functions
The domain of a function is the set of all the first coordinates of the ordered pairs. The range of a function is the set of all the second coordinates.
In the function determined by the grading correspondence in Table 2.1, the domain is the interval [0, 100]. The range is {A, B, C, D, F}.
In a function, each domain element is paired with one and only one range element.
Table 2.1
‹#›
63
Functions
If a function is defined by an equation, the variable that represents elements of the domain is the independent variable. The variable that represents elements of the range is the dependent variable.
For the situation involving the free fall of a rock, we used the equation d = 16t 2.
The elements of the domain represented the time the rock fell, and the elements of the range represented the distance the rock fell.
‹#›
64
Functions
Thus, in d = 16t 2, the independent variable is t and the dependent variable is d.
The specific letters used for the independent and dependent variables are not important.
For example, y = 16x2 represents the same function as d = 16t 2. Traditionally, x is used for the independent variable and y for the dependent variable.
Anytime we use the phrase “y is a function of x” or a similar phrase with different letters, the variable that follows “function of ” is the independent variable.
‹#›
65
Example 1 – Identify Functions
State whether the relation defines y as a function of x.
a. {(2, 3), (4, 1), (4, 5)}
b. 3x + y = 1
c. –4x2 + y2 = 9
d. The correspondence between the x values and the y values in Figure 2.20.
Figure 2.20
‹#›
66
Example 1 – Solution
a. There are two ordered pairs, (4, 1) and (4, 5), with the same first coordinate and different second coordinates.
This set does not define y as a function of x.
b. Solving 3x + y = 1 for y yields y = –3x + 1.
Because –3x + 1 is a unique real number for each x,
this equation defines y as a function of x.
‹#›
67
Example 1 – Solution
c. Solving –4x2 + y2 = 9 for y yields
The right side produces two values of y for each value of x.
For example, when x = 0, y = 3 or y = –3.
Thus –4x2 + y2 = 9 does not define y as a function of x.
cont’d
‹#›
68
Example 1 – Solution
d. Each x is paired with one and only one y.
The correspondence in Figure 2.20 defines y as a function of x.
Figure 2.20
cont’d
‹#›
69
Function Notation
‹#›
70
Function Notation
Functions can be named by using a letter or a combination
of letters, such as f, g, A, log, or tan.
If x is an element of the domain of f, then f (x), which is read “f of x” or “the value of f at x,” is the element in the range of f that corresponds to the domain element x.
‹#›
71
Function Notation
The notation “f ” and the notation “f (x)” mean different things.
“f ” is the name of the function, whereas “f (x)” is the value of the function at x.
Finding the value of f (x) is referred to as evaluating f at x.
To evaluate f (x) at x = a, substitute a for x and simplify.
‹#›
72
Example 2 – Evaluate Functions
Let f (x) = x2 – 1, and evaluate.
a. f (–5)
b. f (3b)
c. 3f (b)
d. f (a + 3)
e. f (a) + f (3)
‹#›
73
Example 2 – Solution
a. f (–5) = (–5)2 – 1
= 25 – 1
= 24
b. f (3b) = (3b)2 – 1
= 9b2 – 1
Substitute –5 for x, and simplify.
Substitute 3b for x, and simplify.
‹#›
74
Example 2 – Solution
c. 3f (b) = 3(b2 – 1)
= 3b2 – 3
d. f (a + 3) = (a + 3)2 – 1
= a2 + 6a + 8
e. f (a) + f (3) = (a2 – 1) + (32 – 1)
= a2 + 7
cont’d
Substitute a + 3 for x.
Simplify.
Substitute a for x; substitute 3 for x.
Simplify.
Substitute b for x, and simplify.
‹#›
75
Function Notation
Sometimes the domain of a function is stated explicitly.
For example, each of f, g, and h below is given by an equation followed by a statement that indicates the domain.
f (x) = x2, x > 0
h(x) = x2, x = 1, 2, 3
Although f and h have the same equation, they are different functions because they have different domains.
‹#›
76
Function Notation
If the domain of a function is not explicitly stated, then its domain is determined by the following convention.
Domain of a Function
Unless otherwise stated, the domain of a function is the set of all real numbers for which the function makes sense and yields real numbers.
‹#›
77
Graphs of Functions
‹#›
78
Graphs of Functions
If a is an element of the domain of a function f, then
(a, f (a)) is an ordered pair that belongs to that function.
Definition of the Graph of a Function
The graph of a function is the graph of all ordered pairs that belong to the function.
‹#›
79
Example 4 – Graph a Function by Plotting Points
Graph each of the following. State the domain of each
function.
a. f (x) = 2x – 3
b. g(x) = 2x2 – 3
c. h(x) = – 3
Solution:
For each part, we create a table of ordered pairs for the function, plot the ordered pairs and then draw a graph through them. Although each of the functions has a similar look, their graphs are quite different.
‹#›
80
Example 4 – Solution
In some cases, it takes a bit of effort to produce, by just plotting points, an accurate graph of a function.
a. Because 2x – 3 is a real number for all values of x, the domain of f is all real numbers.
This can be written
cont’d
‹#›
81
Example 4 – Solution
Plot each ordered pair, and then draw a smooth graph through the points.
The graph is shown in Figure 2.21.
cont’d
Figure 2.21
‹#›
82
Example 4 – Solution
b. Because 2x2 – 3 is a real number for all values of x, the domain of g is all real numbers.
This can be written
cont’d
‹#›
83
Example 4 – Solution
Plot each ordered pair, and then draw a smooth graph through the points. The graph is shown in Figure 2.22.
cont’d
Figure 2.22
‹#›
84
Example 4 – Solution
c. Because is not a real number when x < 0, the domain of h is all real numbers greater than or equal to zero.
This can be written
cont’d
‹#›
85
Example 4 – Solution
Plot each ordered pair, and then draw a smooth graph through the points. The graph is shown in Figure 2.23.
cont’d
Figure 2.23
‹#›
86
Graphs of Functions
Piecewise-defined functions are functions that are represented by more than one expression.
For instance, the function f defined below consists of three pieces, 2x + 1, x2 – 1, and 4 – x.
‹#›
87
Graphs of Functions
To evaluate the function at x, determine the interval in which x lies and then use the expression that corresponds to that interval to evaluate the function.
The following examples show how to evaluate f (–4), f (5),
and f (2).
Since –4 < –2, use 2x + 1. f (–4) = 2(–4) + 1 = –7
Since 5 > 3, use 4 – x. f (5) = 4 – 5 = –1
Since –2 < 2 < 3, use x2 – 1. f (2) = 22 – 1 = 3
‹#›
88
Graphs of Functions
It may be that, for a given number b, there is no number in the domain of f for which f (a) = b.
For instance, suppose f (x) = x2 + 3 and we are asked to find a value in the domain of f for which f (a) = 2.
f (a) = 3
a2 + 3 = 2
a2 = –1
a = i
The values of a are complex numbers and not in the domain of f.
Replace f (a) with a2 + 3.
Solve for a.
‹#›
89
Graphs of Functions
Note from the graph in Figure 2.26 that the horizontal line through (0, 2) does not intersect the graph.
Figure 2.26
‹#›
90
Graphs of Functions
A problem of special interest is determining the values in the domain of a function f for which f (a) = 0.
Zero of a Function
A value a in the domain of a function f for which f (a) = 0 is called a zero of f.
Examples
Let f (x) = 2x – 4. When x = 2, we have, f (x) = 2x – 4
f (2) = 2(2) – 4 = 0.
Because, f (2) = 0, 2 is a zero of f.
‹#›
91
Graphs of Functions
Let g(x) = x2 + 2x – 15. When x = –5 and x = 3, we have
g(x) = x2 + 2x – 15 g(x) = x2 + 2x – 15
g(–5) = (–5)2 + 2(–5) – 15 g(3) = 32 + 2(3) – 15
= 25 – 10 – 15 = 9 + 6 – 15
= 0 = 0
In this case, g(–5) = 0 and g(3) = 0, so there are two zeros of g, –5 and 3.
Let h(x) = x2 + 1. When x = 0, we have
h(x) = x2 + 1
h(0) = 02 + 1
= 1
‹#›
92
Graphs of Functions
In this case, h(0) = 1 0, so 0 is not a zero of the function.
Real Zeros and x-Intercepts Theorem
The real number c is a zero of f if and only if (c, 0) is an x-intercept of the graph of y = f (x).
‹#›
93
Graphs of Functions
The definition of a function as a set of ordered pairs in which no two ordered pairs that have the same first coordinate have different second coordinates implies that any vertical line intersects the graph of a function at no more than one point.
This is known as the vertical line test.
The Vertical Line Test for Functions
A graph is the graph of a function if and only if no vertical line intersects the graph at more than one point.
‹#›
94
Graphs of Functions
Consider the graph in Figure 2.28. As a point on the graph moves from left to right, this graph falls for values of x –2, remains the same height from x = –2 to x = 2, and rises for x 2.
The function represented by the
graph is said to be decreasing
on the interval constant
on the interval [–2, 2], and
increasing on the interval
Figure 2.28
‹#›
95
Graphs of Functions
Definition of Increasing, Decreasing, and Constant Functions
If a and b are elements of an interval I that is a subset of the domain of a function f, then
f is increasing on I if f (a) < f (b) whenever a < b.
f is decreasing on I if f (a) > f (b) whenever a < b.
f is constant on I if f (a) = f (b) for all a and b.
‹#›
96
Graphs of Functions
We know that a function is a relation in which no two ordered pairs that have the same first coordinate have different second coordinates.
This means that, given any x, there is only one y that can be paired with that x .
A function f that satisfies the additional condition that given element b in the range of f there is exactly one element a in the domain of f such that f(a) = b is called a one-to-one function.
‹#›
97
Graphs of Functions
In a manner similar to applying the vertical line test, we can apply a horizontal line test to identify one-to-one functions.
Horizontal Line Test for a One-To-One Function
If every horizontal line intersects the graph of a function at most once, then the graph is the graph of a one-to-one function.
‹#›
98
Graphs of Functions
For example, some horizontal lines intersect the graph in Figure 2.29 at more than one point.
This is not the graph of a one-to-one function.
Figure 2.29
Some horizontal lines intersect this
graph at more than one point. This is
not the graph of a one-to-one function.
‹#›
99
Graphs of Functions
Every horizontal line intersects the graph in Figure 2.30 at most once.
This is the graph of a one-to-one function.
Figure 2.30
Every horizontal line intersects this
graph at most once. This is the graph
of a one-to-one function.
‹#›
100
Copyright © Cengage Learning. All rights reserved.
Linear Functions
SECTION 2.3
‹#›
101
Slopes of Lines
‹#›
102
Slopes of Lines
A function that can be written in the form f (x) = mx + b is called a linear function because its graph is a straight line.
Consider the graph of a straight line in Figure 2.37. Observe that for each 1-unit increase in x, y increases by a constant amount of m units.
Figure 2.37
‹#›
103
Slopes of Lines
In Figure 2.38, note that for each 1-unit increase in x, y decreases by a constant amount of m units.
Graphs of linear functions are characterized by having a constant rise or fall. This rise or fall is called slope.
The graph in Figure 2.37 has a positive slope; the y value is increasing as x increases.
Figure 2.38
‹#›
104
Slopes of Lines
The graph in Figure 2.38 has a negative slope; the y value is decreasing as x increases.
The slope of a line can be calculated by finding the ratio of the change in y between two points to the change in x between the same two points.
For instance, consider the graph of a straight line passing through the points P1(– 4, – 5) and P2(2, 4) shown in Figure 2.39.
Figure 2.39
‹#›
105
Slopes of Lines
The slope of the line between the two points is
The slope of the line is .
‹#›
106
Slopes of Lines
Definition of the Slope of a Nonvertical Line
The slope m of the line passing through the points P1(x1, y1) and P2(x2, y2) with x1 x2 is given by
See Figure 2.40.
Example
The slope of the line through P1(2, –3) and P2(–4, 1) is
Figure 2.40
‹#›
107
Slopes of Lines
When computing the slope of a line, it does not matter which point we label P1 and which we label P2; the value of the slope will be the same.
For instance, if we interchange the two points in the previous example so that we have P1(–4, 1) and P2(2, –3), then
‹#›
108
Slopes of Lines
Frequently, the Greek letter delta () is used to designate
the change in a variable.
Using this notation, y = y2 – y1 and x = x2 – x1.
Using notation, the formula for slope is written .
‹#›
109
Example 1 – Find the Slope of a Line
Find the slope of the line passing through P1 and P2.
a. P1(1, 2), P2(3, 6) b. P1(–3, 4), P2(1, –2)
Solution:
a. The slope of the line passing through P1(1, 2) and P2(3, 6) is
Because m 0, the line slants upward from left to right. The slope of the line is positive. See the graph at the right.
Positive slope
‹#›
110
Example 1 – Solution
b. The slope of the line passing through P1(–3, 4) and P2(1, –2) is
Because m 0, the line slants downward from left to right. The slope of the line is negative. See the graph at the right.
cont’d
Negative slope
‹#›
111
Slopes of Lines
The definition of slope does not apply to vertical lines. Consider, for example, the points P1(3, 7) and P2(3, 2) on the vertical line in Figure 2.41.
Figure 2.41
‹#›
112
Slopes of Lines
Applying the definition of slope to this line produces
Because division by 0 is undefined, we say that the slope of any vertical line is undefined.
The line through P3(1, 5) and P4(4, 5) in Figure 2.41 is a horizontal line. Its slope is given by
The slope of every horizontal line is 0.
Division by 0 is undefined.
‹#›
113
Slopes of Lines
The results for vertical and horizontal lines are summarized below.
Horizontal and Vertical Lines
The graph of x = a is a vertical line through (a, 0). The slope of the line is undefined. See Figure 2.42.
The graph of y = b is a horizontal line through (0, b). The slope of the line is zero. See Figure 2.42.
Figure 2.42
‹#›
114
Slopes of Lines
Example
The graph of x = –2 is a vertical line through (–2, 0). The slope is undefined.
The graph of y = 3 is a horizontal line through (0, 3). The slope is zero.
‹#›
115
Slope–Intercept Form
‹#›
116
Slope–Intercept Form
The graph of f (x) = 2x + 1 is shown in Figure 2.43. Note that the slope of the line between the two points is 2, the coefficient of x in f (x) = 2x + 1. The y-coordinate of the y-intercept is 1, the constant term of f (x) = 2x + 1.
Figure 2.43
‹#›
117
Slope–Intercept Form
Slope–Intercept Form
The graph of f (x) = mx + b is a line with slope m and
y-intercept (0, b).
Example
The graph of f (x) = –2x + 3 is a line with slope –2 and
y-intercept (0, 3).
The graph of f (x) = x – 4 is a line with slope and
y-intercept (0, –4).
‹#›
118
Slope–Intercept Form
If a function is written in the form f (x) = mx + b, then its graph can be drawn by first plotting the y-intercept (0, b) and then using the slope m to determine another point on the line.
‹#›
119
Example 2 – Graph a Linear Function
Graph: f (x) = 2x – 1
Solution:
Replace f (x) with y.
The equation y = 2x – 1 is in slope–intercept form, with b = –1 and m = 2.
Thus the y-intercept is (0, –1) and the slope is 2. Write the slope as
‹#›
120
Example 2 – Solution
To graph the equation, first plot the y-intercept and then use the slope to plot a second point.
This second point is 2 units up (change in y) and 1 unit to the right (change in x) of the y-intercept. See Figure 2.44.
cont’d
y = 2x – 1
Figure 2.44
‹#›
121
Slope–Intercept Form
An equation of the form Ax + By = C, where A, B, and C are real numbers and both A and B are not zero, is called the general form of a linear equation in two variables.
Examples of these equations are given below.
2x – 3y = 6
– 4x + 5y = 0
x = 3
y = –2
A = 2, B = –3, C = 6
A = –4, B = 5, C = 0
A = 1, B = 0, C = 3
A = 0, B = 1, C = –2
‹#›
122
Finding the Equation of a Line
‹#›
123
Finding the Equation of a Line
We can find an equation of a line provided we know its
slope and at least one point on the line. Figure 2.47
suggests that if (x1, y1) is a point on a line l of slope m and
(x, y) is any other point on the line, then
Figure 2.47
‹#›
124
Finding the Equation of a Line
Multiplying each side of the previous equation by x – x1 produces y – y1 = m(x – x1).
This equation is called the point–slope form of the equation of line l.
Point–Slope Form
The graph of
y – y1 = m(x – x1)
is a line that has slope m and passes through (x1, y1).
‹#›
125
Example 4 – Use the Point–Slope Form
Find an equation of the line with slope –3 that passes through (–1, 4).
Solution:
Use the point–slope form with m = –3, x1 = –1, and y1 = 4.
y – y1 = m(x – x1)
y – 4 = –3[x – (–1)]
y – 4 = –3x – 3
y = –3x + 1
Substitute.
Solve for y.
Slope–intercept form
‹#›
126
Parallel and Perpendicular Lines
‹#›