Algebra

1.  Write an equation in standard form of the parabola that has the same shape as the graph of f(x) = 2x2, but with the given point as the vertex (5, 3).

 

A. f(x) = (2x - 4) + 4
B. f(x) = 2(2x + 8) + 3
C. f(x) = 2(x - 5)2 + 3
D. f(x) = 2(x + 3)2 + 3

 

 

2 of 20

5.0 Points

Find the coordinates of the vertex for the parabola defined by the given quadratic function.

 

f(x) = 2(x - 3)2 + 1

 

A. (3, 1)
B. (7, 2)
C. (6, 5)
D. (2, 1)   3 of 20 5.0 Points Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of the following rational function.   g(x) = x + 3/x(x + 4)   A. Vertical asymptotes: x = 4, x = 0; holes at 3x     B. Vertical asymptotes: x = -8, x = 0; holes at x + 4     C. Vertical asymptotes: x = -4, x = 0; no holes     D. Vertical asymptotes: x = 5, x = 0; holes at x - 3               4 of 20 5.0 Points "Y varies directly as the nth power of x" can be modeled by the equation:   A. y = kxn.     B. y = kx/n.     C. y = kx*n.     D. y = knx.               5 of 20 5.0 Points 40 times a number added to the negative square of that number can be expressed as:   A. A(x) = x2 + 20x.     B. A(x) = -x + 30x.     C. A(x) = -x2 - 60x.     D. A(x) = -x2 + 40x.     6 of 20 5.0 Points The graph of f(x) = -x3 __________ to the left and __________ to the right.   A. rises; falls     B. falls; falls     C. falls; rises     D. falls; falls

Solve the following formula for the specified variable:  V = 1/3 lwh for h

7 of 20

Write an equation that expresses each relationship. Then solve the equation for y.  x varies jointly as y and z

A. x = kz; y = x/k
B. x = kyz; y = x/kz
C. x = kzy; y = x/z
D. x = ky/z; y = x/zk

8 of 20

8 times a number subtracted from the squared of that number can be expressed as:

A. P(x) = x + 7x.
B.P(x) = x2 - 8x.
C. P(x) = x - x.
P(x) = x2+ 10x.

9of 20

Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. f(x) = x4 - 9x2

A. x = 0, x = 3, x = -3; f(x) crosses the x-axis at -3 and 3; f(x) touches the x-axis at 0.
B. x = 1, x = 2, x = 3; f(x) crosses the x-axis at 2 and 3; f(x) crosses the x-axis at 0.
C. x = 0, x = -3, x = 5; f(x) touches the x-axis at -3 and 5; f(x) touches the x-axis at 0.
D. x = 1, x = 2, x = -4; f(x) crosses the x-axis at 2 and -4; f(x) touches the x-axis at 0.

10 of 20

Find the domain of the following rational function.   f(x) = x + 7/x2 + 49

A. All real numbers < 69
B. All real numbers > 210
C. All real numbers ≤ 77
D. All real numbers

11 of 20

Write an equation in standard form of the parabola that has the same shape as the graph of f(x) = 3x2 or g(x) = -3x2, but with the given maximum or minimum. Minimum = 0 at x = 11

A. f(x) = 6(x - 9)
B. f(x) = 3(x - 11)2
C. f(x) = 4(x + 10)
D. f(x) = 3(x2 - 15)2

12 of 20

Solve the following polynomial inequality. 3x2 + 10x - 8 ≤ 0

A. [6, 1/3]
B. [-4, 2/3]
C. [-9, 4/5]
D. [8, 2/7]

13 of 20

Find the coordinates of the vertex for the parabola defined by the given quadratic function.  f(x) = -2(x + 1)2 + 5

A. (-1, 5)
B. (2, 10)
C. (1, 10)
D. (-3, 7)

14 of 20 

Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. f(x) = -2x4 + 4x3

A. x = 1, x = 0; f(x) touches the x-axis at 1 and 0
B. x = -1, x = 3; f(x) crosses the x-axis at -1 and 3
C. x = 0, x = 2; f(x) crosses the x-axis at 0 and 2
D. x = 4, x = -3; f(x) crosses the x-axis at 4 and -3

15 of 20

Find the domain of the following rational function. f(x) = 5x/x - 4

A. {x │x ≠ 3}
B. {x │x = 5}
C. {x │x = 2}
D. {x │x ≠ 4}

16 of 20

Based on the synthetic division shown, the equation of the slant asymptote of f(x) = (3x2 - 7x + 5)/x – 4 is:

A. y = 3x + 5.
B. y = 6x + 7.
C. y = 2x - 5.
D. y = 3x2 + 7.

17 of 20

The perimeter of a rectangle is 80 feet. If the length of the rectangle is represented by x, its width can be expressed as:

A. 80 + x.
B. 20 - x.
C. 40 + 4x.
D. 40 - x.

18 of 20

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x) = 2x4 - 4x2 + 1; between -1 and 0

A. f(-1) = -0; f(0) = 2
B. f(-1) = -1; f(0) = 1
C. f(-1) = -2; f(0) = 0
D. f(-1) = -5; f(0) = -3

19 of 20

Solve the following polynomial inequality. 9x2 - 6x + 1 < 0

A. (-∞, -3)
B. (-1, ∞)
C. [2, 4)
D. Ø

20 of 20 

 Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers.  f(x) = x3 - x - 1; between 1 and 2

A. f(1) = -1; f(2) = 5
B. f(1) = -3; f(2) = 7
C. f(1) = -1; f(2) = 3
D. f(1) = 2; f(2) = 7

1. Find the domain of following logarithmic function. f(x) = ln (x - 2)2

A. (∞, 2) ∪ (-2, -∞)
B. (-∞, 2) ∪ (2, ∞)
C. (-∞, 1) ∪ (3, ∞)
D. (2, -∞) ∪ (2, ∞)

2 . Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms to a decimal approximation, of two decimal places, for the solution.

ex = 5.7

A. {ln 5.7}; ≈1.74
B. {ln 8.7}; ≈3.74
C. {ln 6.9}; ≈2.49
D. {ln 8.9}; ≈3.97

3. Evaluate the following expression without using a calculator.

Log7 √7

A. 1/4
B. 3/5
C. 1/2
D. 2/7

4. Write the following equation in its equivalent logarithmic form.

2-4 = 1/16

A. Log4 1/16 = 64
B. Log2 1/24 = -4
C. Log2 1/16 = -4
D. Log4 1/16 = 54

5.  Write the following equation in its equivalent exponential form.

  4 = log2 16

A. 2 log4 = 16
B. 22 = 4
C. 44 = 256
D. 24 = 16

6.  The half-life of the radioactive element krypton-91 is 10 seconds. If 16 grams of krypton-91 are initially present, how many grams are present after 10 seconds? 20 seconds?

A. 10 grams after 10 seconds; 6 grams after 20 seconds
B. 12 grams after 10 seconds; 7 grams after 20 seconds
C. 4 grams after 10 seconds; 1 gram after 20 seconds
D. 8 grams after 10 seconds; 4 grams after 20 seconds

7. Use properties of logarithms to expand the following logarithmic expression as much as possible.

logb (x2y)

A. 2 logy x + logx y
B. 2 logb x + logb y
C. logx - logb y
D. logb x – logx y

8.  Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.

ex+1 = 1/e

A. {-3}
B. {-2}
C. {4}
D. {12}

9. Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. 

log x + 3 log y

A. log (xy)
B. log (xy3)
C. log (xy2)
D. logy (xy)3

10.  Approximate the following using a calculator; round your answer to three decimal places. 

e-0.95

A. .483
B. 1.287
C. .597
D. .387

11. Use the exponential growth model, A = A0ekt, to show that the time it takes a population to double (to grow from A0 to 2A0 ) is given by t = ln 2/k.

A. A0 = A0ekt; ln = ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
B. 2A0 = A0e; 2= ekt; ln = ln ekt; ln 2 = kt; ln 2/k = t
C. 2A0 = A0ekt; 2= ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
D. 2A0 = A0ekt; 2 = ekt; ln 1 = ln ekt; ln 2 = kt; ln 2/k = toe

12. Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.

2 log x = log 25

A. {12}
B. {5}
C. {-3}
D. {25}

13. Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.

31-x = 1/27

A. {2}
B. {-7}
C. {4}
D. {3}

14.Find the domain of following logarithmic function.

f(x) = log (2 - x)

A. (∞, 4)
B. (∞, -12)
C. (-∞, 2)
D. (-∞, -3)

15.Use properties of logarithms to expand the following logarithmic expression as much as possible.

  Logb (√xy3 / z3)

A. 1/2 logb x - 6 logb y + 3 logb z
B. 1/2 logb x - 9 logb y - 3 logb z
C. 1/2 logb x + 3 logb y + 6 logb z
D. 1/2 logb x + 3 logb y - 3 logb z

16.    Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms to a decimal approximation, of two decimal places, for the solution.

32x + 3x - 2 = 0

A. {1}
B. {-2}
C. {5}
D. {0}

17.   You have $10,000 to invest. One bank pays 5% interest compounded quarterly and a second bank pays 4.5% interest compounded monthly. Use the formula for compound interest to write a function for the balance in each bank at any time t.

A. A = 20,000(1 + (0.06/4))4t; A = 10,000(1 + (0.044/14))12t
B. A = 15,000(1 + (0.07/4))4t; A = 10,000(1 + (0.025/12))12t
C. A = 10,000(1 + (0.05/4))4t; A = 10,000(1 + (0.045/12))12t
D. A = 25,000(1 + (0.05/4))4t; A = 10,000(1 + (0.032/14))12t

18.  The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.

A. bx; (∞, -∞); (1, ∞)
B. bx; (-∞, -∞); (2, ∞)
C. bx; (-∞, ∞); (0, ∞)
D. bx; (-∞, -∞); (-1, ∞)

19. 

Write the following equation in its equivalent exponential form.   log6 216 = y   A. 6y = 216   B. 6x = 216   C. 6logy = 224   D. 6xy = 232

20. Write the following equation in its equivalent exponential form.

  5 = logb 32

A. b5 = 32
B. y5 = 32
C. Blog5 = 32
D. Logb = 32

21. Perform the long division and write the partial fraction decomposition of the remainder term. 

x5 + 2/x2 - 1

A. x2 + x - 1/2(x + 1) + 4/2(x - 1)
B. x3 + x - 1/2(x + 1) + 3/2(x - 1)
C. x3 + x - 1/6(x - 2) + 3/2(x + 1)
D. x2 + x - 1/2(x + 1) + 4/2(x - 1)

22.  Solve each equation by the substitution method.

x + y = 1  x2 + xy – y2 = -5

A. {(4, -3), (-1, 2)}
B. {(2, -3), (-1, 6)}
C. {(-4, -3), (-1, 3)}
D. {(2, -3), (-1, -2)}

23.   A television manufacturer makes rear-projection and plasma televisions. The profit per unit is $125 for the rear-projection televisions and $200 for the plasma televisions. 

Let x = the number of rear-projection televisions manufactured in a month and let y = the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit.

A. z = 200x + 125y
B. z = 125x + 200y
C. z = 130x + 225y
D. z = -125x + 200y

24. Solve the following system by the substitution method.

{x + 3y = 8  {y = 2x - 9

A. {(5, 1)}
B. {(4, 3)}
C. {(7, 2)}
D. {(4, 3)}

 

25. Solve the following system by the addition method.

{4x + 3y = 15  {2x – 5y = 1

A. {(4, 0)}
B. {(2, 1)}
C. {(6, 1)}
D. {(3, 1)}

26. Many elevators have a capacity of 2000 pounds. 

If a child averages 50 pounds and an adult 150 pounds, write an inequality that describes when x children and y adults will cause the elevator to be overloaded.

A. 50x + 150y > 2000
B. 100x + 150y > 1000
C. 70x + 250y > 2000
D. 55x + 150y > 3000

27.Find the quadratic function y = ax2 + bx + c whose graph passes through the given points.

(-1, -4), (1, -2), (2, 5)

A. y = 2x2 + x - 6
B. y = 2x2 + 2x - 4
C. y = 2x2 + 2x + 3
D. y = 2x2 + x - 5

28.Write the partial fraction decomposition for the following rational expression.

  x + 4/x2(x + 4)

A. 1/3x + 1/x2 - x + 5/4(x2 + 4)
B. 1/5x + 1/x2 - x + 4/4(x2 + 6)
C. 1/4x + 1/x2 - x + 4/4(x2 + 4)
D. 1/3x + 1/x2 - x + 3/4(x2 + 5)

29.  Find the quadratic function y = ax2 + bx + c whose graph passes through the given points.

(-1, 6), (1, 4), (2, 9)

A. y = 2x2 - x + 3
B. y = 2x2 + x2 + 9
C. y = 3x2 - x - 4
D. y = 2x2 + 2x + 4

30. Solve the following system.

3(2x+y) + 5z = -1  2(x - 3y + 4z) = -9  4(1 + x) = -3(z - 3y)

A. {(1, 1/3, 0)}
B. {(1/4, 1/3, -2)}
C. {(1/3, 1/5, -1)}
D. {(1/2, 1/3, -1)}

31. On your next vacation, you will divide lodging between large resorts and small inns. Let x represent the number of nights spent in large resorts. Let y represent the number of nights spent in small inns. 

Write a system of inequalities that models the following conditions:  You want to stay at least 5 nights. At least one night should be spent at a large resort. Large resorts average $200 per night and small inns average $100 per night. Your budget permits no more than $700 for lodging.

A. y ≥ 1  x + y ≥ 5 x ≥ 1  300x + 200y ≤ 700
B. y ≥ 0 x + y ≥ 3  x ≥ 0  200x + 200y ≤ 700
C. y ≥ 1 x + y ≥ 4 x ≥ 2  500x + 100y ≤ 700
D. y ≥ 0 x + y ≥ 5 x ≥ 1  200x + 100y ≤ 700

32. Solve the following system.

2x + 4y + 3z = 2  x + 2y - z = 0  4x + y - z = 6

A. {(-3, 2, 6)}
B. {(4, 8, -3)}
C. {(3, 1, 5)}
D. {(1, 4, -1)}

33. Solve the following system.

x = y + 4  3x + 7y = -18

A. {(2, -1)}
B. {(1, 4)}
C. {(2, -5)}
D. {(1, -3)}

34.  Write the partial fraction decomposition for the following rational expression.

4/2x2 - 5x – 3

A. 4/6(x - 2) - 8/7(4x + 1)
B. 4/7(x - 3) - 8/7(2x + 1)
C. 4/7(x - 2) - 8/7(3x + 1)
D. 4/6(x - 2) - 8/7(3x + 1)

35.  Solve each equation by either substitution or addition method.

x2 + 4y2 = 20  x + 2y = 6

A. {(5, 2), (-4, 1)}
B. {(4, 2), (3, 1)}
C. {(2, 2), (4, 1)}
D. {(6, 2), (7, 1)}

36.  Solve each equation by either substitution or addition method.

x2 + 4y2 = 20  x + 2y = 6

A. {(5, 2), (-4, 1)}
B. {(4, 2), (3, 1)}
C. {(2, 2), (4, 1)}
D. {(6, 2), (7, 1)}

37.  Write the partial fraction decomposition for the following rational expression.

1/x2 – c2 (c ≠ 0)

A. 1/4c/x - c - 1/2c/x + c
B. 1/2c/x - c - 1/2c/x + c
C. 1/3c/x - c - 1/2c/x + c
D. 1/2c/x - c - 1/3c/x + c

38. Solve each equation by the substitution method.

x2 - 4y2 = -7  3x2 + y2 = 31

A. {(2, 2), (3, -2), (-1, 2), (-4, -2)}
B. {(7, 2), (3, -2), (-4, 2), (-3, -1)}
C. {(4, 2), (3, -2), (-5, 2), (-2, -2)}
D. {(3, 2), (3, -2), (-3, 2), (-3, -2)}

39. Solve each equation by the substitution method.

y2 = x2 - 9  2y = x – 3

A. {(-6, -4), (2, 0)}
B. {(-4, -4), (1, 0)}
C. {(-3, -4), (2, 0)}
D. {(-5, -4), (3, 0)}

40. Write the partial fraction decomposition for the following rational expression.

  ax +b/(x – c)2 (c ≠ 0)

A. a/a – c +ac + b/(x – c)2
B. a/b – c +ac + b/(x – c)
C. a/a – b +ac + c/(x – c)2
D. a/a – b +ac + b/(x – c)