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Precalculus Spring Final Exam Review Questions 2013 Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. The graph of a logarithmic function is shown. Select the function which matches the graph. a. y = log x - 2 b. y = 2 - log x ____ 2. The graph of a logarithmic function is shown. Select the function which matches the graph. a. y = log3(-x) b. y = log3x ____ c. y = log(x - 2) d. y = log(2 - x) c. y = -log3x d. y = 1 - log3x 3. Match the equation to its graph. y2 = -12x a. c. b. ____ d. 4. Match the equation to the graph. (y + 2)2 = 6(x + 2) a. c. b. d. ____ 5. Match the graph to its equation. a. b. ____ - =1 - =1 c. d. 6. Match the equation to the graph. a. = 1 c. + =1 + =1 b. d. Short Answer 7. Find the x- and y-intercepts of f. f(x) = -x2(x + 4)(x2 + 1) 8. Use the graph to find the vertical asymptotes, if any, of the function. 9. Use the graph to find the horizontal asymptote, if any, of the function. 10. Give the equation of the horizontal asymptote, if any, of the function. g(x) = 11. Use the Remainder Theorem to find the remainder when f(x) is divided by x - c. f(x) = 5x6 - 3x3 + 8; x + 1 12. Find the real solutions of the equation. 3x4 - 17x3 + 61x2 - 73x + 26 = 0 13. Find a bound on the real zeros of the polynomial function. x5 + 2x4 + 2x3 - 7x2 + x + 4 14. Change the exponential expression to an equivalent expression involving a logarithm. ex = 6 15. Find the domain of the function. f(x) = ln 16. Suppose that ln 2 = a and ln 5 = b. Use properties of logarithms to write each logarithm in terms of a and b. ln 17. Solve the problem. The formula A = 165e0.039t models the population of a particular city, in thousands, t years after 1998. When will the population of the city reach 296 thousand? 18. Find the present value. Round to the nearest cent. To get $2000 after 7 years at 9% compounded semiannually 19. Solve the problem. The half-life of plutonium-234 is 9 hours. If 60 milligrams is present now, how much will be present in 2 days? (Round your answer to three decimal places.) 20. Solve the problem. The minute hand of a clock is 3 inches long. How far does the tip of the minute hand move in 10 minutes? If necessary, round the answer to two decimal places. 21. If A denotes the area of the sector of a circle of radius r formed by the central angle , find the missing quantity. If necessary, round the answer to two decimal places. r = 44.2 centimeters, = radians, A = ? 22. In the problem, t is a real number and P = (x, y) is the point on the unit circle that corresponds to t. Find the exact value of the indicated trigonometric function of t. ( , ) Find sin t. 23. Solve the problem. What is the domain of the cosine function? 24. Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a calculator. tan 390° 25. Name the quadrant in which the angle lies. csc > 0, sec > 0 26. Find the exact value of the indicated trigonometric function of . tan = , 180°< < 270° Find cos . 27. Find the phase shift. y = -5 cos(8x + ) 28. Find the equation of the parabola described. Focus at (5, 0); vertex at (0, 0) 29. Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola. - =1 30. Form a polynomial whose zeros and degree are given. Zeros: -5, -4, 1, 2; degree 4 31. For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. f(x) = 5(x2 + 3)(x + 2)2 32. Find the domain of the rational function. G(x) = 33. Give the equation of the horizontal asymptote, if any, of the function. Q(x) = 34. Give the equation of the oblique asymptote, if any, of the function. Q(x) = 35. Find the domain of the rational function. g(x) = 36. Find the indicated intercept(s) of the graph of the function. y-intercept of f(x) = 37. Use the Factor Theorem to determine whether x - c is a factor of f. If it is, write f in factored form, that is, write f in the form f(x) = (x - c)(quotient). f(x) = 5x4 - 8x3 + 18x2 - 24x + 9; x - 1 38. List the potential rational zeros of the polynomial function. Do not find the zeros. f(x) = x5 - 5x2 + 4x + 3 39. Use the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. f(x) = x4 - 3x2 - 4 40. Find the real solutions of the equation. x4 - 5x2 - 36 = 0 41. Solve the problem. One solution of x3 - 5x2 + 5x - 1 = 0 is 1. Find the other two solutions. 42. Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 5; zeros: 6, 3 + 5i, -6i 43. Find all zeros of the function and write the polynomial as a product of linear factors. f(x) = x4 + 25x2 + 144 44. Indicate whether the function is one-to-one. {(6, 1), (-3, 2), (-5, 3), (-7, 4)} 45. Decide whether or not the functions are inverses of each other. f(x) = 5x - 5, g(x) = x+1 46. The function f is one-to-one. Find its inverse. f(x) = (x + 2)3 - 8. 47. Determine the exponential function whose graph is given. 48. Solve the problem. The rabbit population in a forest area grows at the rate of 9% monthly. If there are 270 rabbits in July, find how many rabbits (rounded to the nearest whole number) should be expected by next July. Use 49. Change the exponential expression to an equivalent expression involving a logarithm. 72 = 49 50. Change the logarithmic expression to an equivalent expression involving an exponent. log1/416 = -2 51. Find the exact value of the logarithmic expression. ln e 52. Find the domain of the function. f(x) = log10 53. Graph the function. f(x) = log5x 54. Solve the problem. The pH of a solution ranges from 0 to 14. An acid has a pH less than 7. Pure water is neutral and has a pH of 7. The pH of a solution is given by where x represents the concentration of the hydrogen ions in the solution in moles per liter. Find the hydrogen ion concentration if the 55. Solve the equation. 4(7 + 3x) = 56. Solve the problem. Randy invested his inheritance in an account that paid 6.2% interest, compounded continuously. After 7 years, he found that he now had $55,919.61. What was the original amount of his inheritance? 57. Solve the problem. How long will it take a sample of radioactive substance to decay to half of its original amount, if it decays according to the function A(t) = 550e-0.158t, where t is the time in years? Round to the nearest hundredth year. 58. Convert the angle in degrees to radians. Express the answer as multiple of . -105° 59. In the problem, t is a real number and P = (x, y) is the point on the unit circle that corresponds to t. Find the exact value of the indicated trigonometric function of t. (- , ) Find cot t. 60. Find the exact value of the indicated trigonometric function of . tan = - , in quadrant II Find cos . 61. Use the even-odd properties to find the exact value of the expression. Do not use a calculator. csc (-60°) 62. Without graphing the function, determine its amplitude or period as requested. y = cos(3x) Find the period. 63. Write the equation of a sine function that has the given characteristics. Amplitude: 3 Period: 4 64. Find an equation for the graph. 65. Find the phase shift. y = -4 cos 66. Find the phase shift. y = 5 cos 67. Graph the function. Show at least one period. y = 5 sin(4x - ) 68. Graph the function. Show at least one period. y = -4 cos(x + 5) 69. Find the equation of the parabola described. Vertex at (0, 0); axis of symmetry the x-axis; containing the point (4, 3) 70. Find the vertex, focus, and directrix of the parabola. Graph the equation. x2 = -16y 71. Find an equation for the ellipse described. Vertices at (-2, 2) and (12, 2); focus at (10, 2) 72. Find an equation for the ellipse. Foci at (0, ±3); a= 4 Precalculus Spring Final Exam Review Questions 2013 Answer Section MULTIPLE CHOICE 1. 2. 3. 4. 5. 6. ANS: ANS: ANS: ANS: ANS: ANS: A D A B C A PTS: PTS: PTS: PTS: PTS: PTS: 1 1 1 1 1 1 SHORT ANSWER 7. ANS: x-intercepts: -4, 0; y-intercept: 0 PTS: 1 8. ANS: x = -2, x = 2 PTS: 1 9. ANS: y=3 PTS: 1 10. ANS: none PTS: 1 11. ANS: 16 PTS: 1 12. ANS: PTS: 1 13. ANS: -8 and 8 PTS: 1 14. ANS: ln 6 = x PTS: 1 15. ANS: (0, ) PTS: 1 16. ANS: (3a + b) PTS: 1 17. ANS: 2013 PTS: 1 18. ANS: $1079.95 PTS: 1 19. ANS: 1.488 PTS: 1 20. ANS: 3.14 in. PTS: 1 21. ANS: 383.6 cm2 PTS: 1 22. ANS: PTS: 1 23. ANS: all real numbers PTS: 1 24. ANS: PTS: 1 25. ANS: I PTS: 1 26. ANS: PTS: 1 27. ANS: units to the left PTS: 1 28. ANS: y2 = 20x PTS: 1 29. ANS: center at (2, 3) transverse axis is parallel to x-axis vertices at (-3, 3) and (7, 3) foci at (2 , 3) and (2 + , 3) asymptotes of y - 3 = - (x - 2) and y - 3 = PTS: 1 30. ANS: x4 + 6x3 - 5x2 - 42x + 40 PTS: 1 31. ANS: -2, multiplicity 2, touches x-axis PTS: 1 32. ANS: {x|x -5, x PTS: 1 33. ANS: y = -1 PTS: 1 34. ANS: y=x+6 PTS: 1 35. ANS: {x|x 1} PTS: 1 36. ANS: 6} (x - 2) PTS: 1 37. ANS: Yes; f(x) = (x - 1)(5x3 - 3x2 + 15x - 9) PTS: 1 38. ANS: ± 1, ± 3 PTS: 1 39. ANS: -2, 2; f(x) = (x - 2)(x + 2)(x2 + 1) PTS: 1 40. ANS: {-3, 3} PTS: 1 41. ANS: {2 + ,2- } PTS: 1 42. ANS: 3 - 5i, 6i PTS: 1 43. ANS: f(x) = (x + 3i)(x - 3i)(x + 4i)(x - 4i) PTS: 1 44. ANS: Yes PTS: 1 45. ANS: Yes PTS: 1 46. ANS: f-1(x) = PTS: 1 47. ANS: f(x) = 2x PTS: 1 -2 48. ANS: 789 PTS: 1 49. ANS: log749 = 2 PTS: 1 50. ANS: -2 = 16 PTS: 1 51. ANS: 1 PTS: 1 52. ANS: (-, -8) (2, ) PTS: 1 53. ANS: PTS: 1 54. ANS: 3.98 10-7 PTS: 1 55. ANS: {-3} PTS: 1 56. ANS: $36,231 PTS: 1 57. ANS: 4.39 yr PTS: 1 58. ANS: PTS: 1 59. ANS: PTS: 1 60. ANS: PTS: 1 61. ANS: 2 PTS: 1 62. ANS: PTS: 1 63. ANS: y = 3 sin PTS: 1 64. ANS: y = 2 cos PTS: 1 65. ANS: units to the left PTS: 1 66. ANS: units to the left PTS: 1 67. ANS: PTS: 1 68. ANS: PTS: 1 69. ANS: y2 = x PTS: 1 70. ANS: vertex: (0, 0) focus: (0, -4) directrix: y = 4 PTS: 1 71. ANS: + =1 PTS: 1 72. ANS: + PTS: 1 =1