# Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around

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1. 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

2. Determine the degree and the leading coefficient of the polynomial function f(x) = -2x3 (x - 1)(x + 5).

A. 5; -2

B. 7; -4

C. 2; -5

D. 1; -9

3. 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}

4. 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

5. "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.

6. 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

7. 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)

8. 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) = x3 + 2x2 - x - 2

A. x = 2, x = 2, x = -1; f(x) touches the x-axis at each.

B. x = -2, x = 2, x = -5; f(x) crosses the x-axis at each.

C. x = -3, x = -4, x = 1; f(x) touches the x-axis at each.

D. x = -2, x = 1, x = -1; f(x) crosses the x-axis at each.

9. 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

10. The difference between two numbers is 8. If one number is represented by x, the other number can be expressed as:

A. x - 5.

B. x + 4.

C. x - 8.

D. x - x.

11. Solve the following polynomial inequality.

9x2 - 6x + 1 < 0

A. (-∞, -3)

B. (-1, ∞)

C. [2, 4)

D. Ø

12. 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)

13. 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

14. If f is a polynomial function of degree n, then the graph of f has at most __________ turning points.

A. n - 3

B. n - f

C. n - 1

D. n + f

15. All rational functions can be expressed as f(x) = p(x)/q(x), where p and q are __________ functions and q(x) ≠ 0.

A. horizontal asymptotes

B. polynomial

C. vertical asymptotes

D. slant asymptotes

16. 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.

17. Find the domain of the following rational function.

g(x) = 3x2/((x - 5)(x + 4))

A. {x│ x ≠ 3, x ≠ 4}

B. {x│ x ≠ 4, x ≠ -4}

C. {x│ x ≠ 5, x ≠ -4}

D. {x│ x ≠ -3, x ≠ 4}

18. 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.

Maximum = 4 at x = -2

A. f(x) = 4(x + 6)2 - 4

B. f(x) = -5(x + 8)2 + 1

C. f(x) = 3(x + 7)2 - 7

D. f(x) = -3(x + 2)2 + 4

19. 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.

D.

P(x) = x2+ 10x.

20. 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

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

log2 96 – log2 3

A. 5

B. 7

C. 12

D. 4

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

logb (x2 y) / z2

A. 2 logb x + logb y - 2 logb z

B. 4 logb x - logb y - 2 logb z

C. 2 logb x + 2 logb y + 2 logb z

D. logb x - logb y + 2 logb z

23. 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, ∞)

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

3 ln x – 1/3 ln y

A. ln (x / y1/2)

B. lnx (x6 / y1/3)

C. ln (x3 / y1/3)

D. ln (x-3 / y1/4)

25. 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

26. 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}

27. The graph of the exponential function f with base b approaches, but does not touch, the __________-axis. This axis, whose equation is __________, is a __________ asymptote.

A. x; y = 0; horizontal

B. x; y = 1; vertical

C. -x; y = 0; horizontal

D. x; y = -1; vertical

28. 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

29. Consider the model for exponential growth or decay given by A = A0ekt. If k __________, the function models the amount, or size, of a growing entity. If k __________, the function models the amount, or size, of a decaying entity.

A. > 0; < 0

B. = 0; ≠ 0

C. ≥ 0; < 0

D. < 0; ≤ 0

30. 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

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

5 = logb 32

A. b5 = 32

B. y5 = 32

C. Blog5 = 32

D. Logb = 32

32. 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}

33. 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

34. Find the domain of following logarithmic function.

f(x) = log5 (x + 4)

A. (-4, ∞)

B. (-5, -∞)

C. (7, -∞)

D. (-9, ∞)

35. 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}

36. Evaluate the following expression without using a calculator.

Log7 √7

A. 1/4

B. 3/5

C. 1/2

D. 2/7

37. 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, ∞)

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

log6 216 = y

A.

6y = 216

B.

6x = 216

C.

6logy = 224

D.

6xy = 232

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

3√8 = 2

A. Log2 3 = 1/8

B. Log8 2 = 1/3

C. Log2 8 = 1/2

D. Log3 2 = 1/8

40. Evaluate the following expression without using a calculator.

8log8 19

A. 17

B. 38

C. 24

D. 19

41. 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)}

42. Solve the following system.

 x = y + 4 3x + 7y = -18

A. {(2, -1)}

B. {(1, 4)}

C. {(2, -5)}

D. {(1, -3)}

43. 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)}

44. 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

45. 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)}

46. 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

47. Solve each equation by the addition method.

 x2 + y2 = 25 (x - 8)2 + y2 = 41

A. {(3, 5), (3, -2)}

B. {(3, 4), (3, -4)}

C. {(2, 4), (1, -4)}

D. {(3, 6), (3, -7)}

48. 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)}

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

x4 – x2 + 2/x3 - x2

A. x + 3 - 2/x - 1/x2 + 4x - 1

B. 2x + 1 - 2/x - 2/x + 2/x + 1

C. 2x + 1 - 2/x2 - 2/x + 5/x - 1

D. x + 1 - 2/x - 2/x2 + 2/x - 1

50. 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)

51. 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)

52. 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)}

53. 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

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

x2 – 6x + 3/(x – 2)3

A. 1/x – 4 – 2/(x – 2)2 – 6/(x – 2)

B. 1/x – 2 – 4/(x – 2)2 – 5/(x – 1)3

C. 1/x – 3 – 2/(x – 3)2 – 5/(x – 2)

D. 1/x – 2 – 2/(x – 2)2 – 5/(x – 2)3

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

6x - 11/(x - 1)2

A. 6/x - 1 - 5/(x - 1)2

B. 5/x - 1 - 4/(x - 1)2

C. 2/x - 1 - 7/(x - 1)

D. 4/x - 1 - 3/(x - 1)

56. Write the form of the partial fraction decomposition of the rational expression.
7x - 4/x2 - x - 12

A. 24/7(x - 2) + 26/7(x + 5)

B. 14/7(x - 3) + 20/7(x2 + 3)

C. 24/7(x - 4) + 25/7(x + 3)

D. 22/8(x - 2) + 25/6(x + 4)

57. 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)}

58. Solve the following system.

 x + y + z = 6 3x + 4y - 7z = 1 2x - y + 3z = 5

A. {(1, 3, 2)}

B. {(1, 4, 5)}

C. {(1, 2, 1)}

D. {(1, 5, 7)}

59. 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)}

60. 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

Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept
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