AP Precalculus Semester Exam Review PDF

# AP Precalculus - Semester Exam Review ## Part 1 **1) What are the intervals of increasing and decreasing of f(x) = (x - 3)^2 - 4** * Increasing: (3, ∞) * Decreasing: (-∞, 3) * Positive: (-∞, 1) U (5, ∞) * Negative: (1, 5) **2) Evaluate f(-3) if f(x) = 4x^2 - 3x + 2** f(-3) = 4(-3)^2 - 3(-3) + 2...

# AP Precalculus - Semester Exam Review ## Part 1 **1) What are the intervals of increasing and decreasing of f(x) = (x - 3)^2 - 4** * Increasing: (3, ∞) * Decreasing: (-∞, 3) * Positive: (-∞, 1) U (5, ∞) * Negative: (1, 5) **2) Evaluate f(-3) if f(x) = 4x^2 - 3x + 2** f(-3) = 4(-3)^2 - 3(-3) + 2 = 47 **3) Find the average rate of change over the interval [-2, 4] if f(x) = x^2 - 1** * f(-2) = 3 * f(4) = 15 Average Rate of Change = (15 - 3) / (4 - (-2)) = 2 **4) Find the average rate of change from x₁ = 2 to x₂ = 7. f(x) = x^2 - 3x** * f(2) = -2 * f(7) = 28 Average Rate of Change = (28 - (-2)) / (7 - 2) = 10 **5) Write the equation of the secant line containing the two points from problem 4.** * m = 6 * (2, -2) y + 2 = 6(x - 2) y = 6x - 14 **6) Complete the following table. y = 4 - x^2** | Interval | Average Rate of Change | |---|---| | [-1, 0] | -1 | | [0, 1] | -2 | | [1, 2] | -3 | | [2, 3] | -5 | **7) What is the difference in the average rates of change?** 2 **8) Where is the rate of change constant?** At the vertex; At x=0 **9) Write the equation that represents the average rate of change.** y = -2x ## Part 2 **10) Find the extrema (Round to the nearest thousandth, if necessary). y = x^4 - 4x^3 + 3x** * Maximum: (0.554, 1.076) * Minimum: (-0.465, -0.946) * Minimum: (2.912, -18.13) **11) Write a polynomial with real coefficients having degree 4 and zeros -3i and 4 (multiplicity 2).** p(x) = (x - 3i)(x + 3i)(x - 4)² = x^4 − 8x^3 + 25x^2 − 72x + 144 **For problems 12-14, determine if the function is even, odd, or neither.** **12) f(x) = 3x^2 - 2x^4** f(-x) = 3(-x)^2 - 2(-x)^4 = f(x) Even **13) f(x) = 2x^5 + 3x^3 + x** f(-x) = -2x^5 - 3x^3 - x = -f(x) Odd **14) f(x) = 2x^2 + 3x + 1** Neither **For problems 15 and 16, solve using sign analysis.** **15) x^3 + 5x^2 - 9x < 45** (x + 3)(x - 3)(x + 5) < 0 * Solution: (-10, -5) U (-3, 3) **16) x^3 - 3x^2 - 4x ≥ -12** (x + 2)(x - 2)(x - 3) ≥ 0 * Solution: [-2, 2] U [3, ∞) **Use the following function for problems 17 – 19, f(x) = (x – 2)^2(x + 3)** **17) Identify the zeros and state the multiplicities.** * x = 2, multiplicity of 2 * x = -3, multiplicity of 1 **18) State the end behavior using limit notation.** * lim(x→-∞) f(x) = -∞ * lim(x→∞) f(x) = ∞ **19) Sketch a possible graph using the zeros, multiplicities, and end behavior.** * The graph has a zero at x = -3 with multiplicity 1, so it crosses the x-axis at that point. * The graph has a zero at x = 2 with multiplicity 2, so it touches the x-axis at that point and doesn't cross. * The end behavior indicates that the graph goes down as x approaches negative infinity and up as x approaches positive infinity. **Use the following function for problems 20 – 26, f(x) = ((3x ^2 + 5x - 2)) / (x^2 - 4)** **20) Write the domain.** (-∞, -2) U (-2, 2) U (2, ∞) **21) Write the range.** (-∞, 7/4) U (7/4, 3) U (3, ∞) **22) Find the vertical asymptote.** x = 2 **23) Find the horizontal asymptote.** y = 3 **24) Find the zeros.** x = 1/3 **25) Find the coordinates of the hole.** (-2, 7/4) **26) Write the end behavior in limit notation.** * lim(x→-∞) f(x) = 3 * lim(x→∞) f(x) = 3 **27) Use the binomial theorem to expand (2x - 1)^4** (2x)^4 + 4(2x)^3(-1) + 6(2x)^2(-1)^2 + 4(2x)(-1)^3 + 1(-1)^4= 16x^4 − 32x^3 + 24x^2 − 8x + 1 **28) Solve the inequality using sign analysis. (2-x) / (x+3) > 0** * Solution: (-3, 2) **29) Find a linear model and use it to make a prediction. The value of a copy machine is $10,325 when purchased. After two years its value is $8,375. What is the value after 8 years?** y = -975x + 10325 * Value after 8 years: $2525 **Use the table for problems 30 and 31.** | Year | Cost | |---|---| | 2010 | 10.35 | | 2011 | 10.80 | | 2012 | 11.00 | | 2013 | 11.90| | 2014 | 12.25| **30) Let x = 0 be 2010, find a linear regression to model the situation.** y = 0.49x + 10.28 **31) Use the equation to predict the cost in 2019.** y(9) = 14.69 ## Part 3 **Use the following information for problems 32 – 34.** A square of side x-inches is cut out of each corner of an 8 in by 11 in piece of cardboard and the sides are folded up to form an open-topped box. **32) Write the value of V as a function of x.** V(x) = x(8 - 2x)(11 - 2x) **33) State the domain of the function.** 0 < x < 4 **34) Graph the function to find the maximum volume of the box. What is the maximum volume and what value of x gives the maximum volume?** * Maximum volume: 60,013 in^3 * x ≈ 1.525 in **Use the following information for problems 35 - 37.** The number of elk P at any time t (in years) in a national game reserve is given by P(t) = (500 + 624t) / (20 + 0.8t) **35) Find the number of elk when t = 25.** P(25) = 402.5 **36) Find the horizontal asymptote.** y = 780 **37) According to the model, what is the largest possible elk population?** 780 elk ## Part 4 **38) Write the general rule for the arithmetic sequence with a₁ = 20 and d = 3.** a_n = 20 + 3(n - 1) **39) Find the 31st term of the sequence 22, 19, 16...** a_n = 22 - 3(n - 1) a_31 = -68 **40) Write the explicit formula for the geometric sequence with a₁ = 12 and r = -1/2.** a_n = 12(-1/2)^(n - 1) **41) Find the 10th term of the sequence 1/3, 1, 3, 9...** a_n = (1/3)(3)^(n-1) a_10 = 6,561 **42) Given the arithmetic sequence where a3 = -6 and a9 = 18, write a linear function that will contain the domain of the sequence.** y = 4x - 18 **43) Given the geometric sequence where a3 = 2 and r = 3, write an exponential function that will contain the domain of the sequence.** f(n) = 2(3)^(n - 3) **44) A culture contains 3000 bacteria initially and increases by 30% every hour. Find a formula for the number N(t) of bacteria present after t hours. How many bacteria are in the culture after 8 hours?** N(t) = 3000(1.3)^t N(8) = 24,471.9 **For problems 45 and 46, determine whether the equation represents exponential growth or exponential decay. Explain** **45) f(x) = 312(1.325)^x** Growth because the base is greater than 1. **46) f(x) = 19.5(0.937)^x** Decay because the base is less than 1. **47) Describe the end behavior of f(x) = -3(2)^x + 1 using limit notation.** * lim(x→-∞) f(x) = 1 * lim(x→∞) f(x) = -∞ **For problems 48 and 49, describe the transformations of f that yield the graph of g.** **48) f(x) = 2^x; g(x) = -2^x + 3** * Reflected over the x-axis * Shifted up by 3 units. **49) f(x) = 3^x; g(x) = 3^(x−5) + 2** * Shifted 5 units to the right. * Shifted 2 units up. **50) List the domain and range of g(x) = 2^(x-3) - 4** * Domain: (-∞, ∞) * Range: (-4, ∞). **51) A car sells for $23,500. It depreciates 18% every year. Write an exponential function to model the situation. What is the value of the car after 4 years?** A(t) = 23500(0.82)^t A(4) = $10,624.86 **52) Bacteria is being grown in a Petri dish. The number of bacteria B after t hours can be modeled by B(t) = 140e^(0.535t). What is the initial number of bacteria and how many are present after 10 hours?** * Initial number of bacteria: 140 * Number of bacteria after 10 hours: 29,485 **53) The graph of f is shown in the image at right. Write the equation for the secant line on the interval -2 ≤ x ≤ 4 .** y = -2x + 3 **Use the tables for 63 and 64.** | x | f(x) | |---|---| | -1 | 3 | | 0 | 2 | | 1 | -1 | | 2 | 0 | | 3 | 5 | | x | g(x) | |---|---| | -1 | 0 | | -2 | -3 | | 2 | 2 | | 4 | 1 | | 5 | -1 | | 6 | -1 | **63) Find f^-1 (5)** f(5) = 3 **64) Find g^-1 (f(0))** g^-1 (f(0)) = g^-1(2) = 4 **Use the following information for problems 65 and 66.** The formula for the volume of a cylinder whose height is equal to its diameter is V = 2πr^3 **65) Use inverse operations to solve for r in terms of V.** r = (3√(V / 2π)) **66) Use the equation to find the radius if the volume is 90 cubic inches.** r ≈ 2.429 in **67) The table shows values for a function g at selected values of x. Which of the following statements best fit these data?** | x | g(x) | |---|---| | 1 | 20 | | 3 | 13 | | 5 | 8 | | 7 | 5 | | 9 | 4 | * g is best modeled by a quadratic function because the change in the average rate of change is constant. **55) If f(x) = x^2 + 2 and g(x) = x - 5, find f o g and state its domain.** f(g(x)) = x^2 - 10x + 27 * Domain: (-∞, ∞) **Use the table for problems 56 – 58.** | x | f(x) | g(x) | |---|---|---| | -3 | 2 | 0 | | -2 | -1 | -3 | | -1 | 0 | 1 | | 0 | 1 | 2 | | 1 | -3 | -1 | | 2 | -2 | 0 | **56) Find f(g(-2))** f(g(-2)) = 2 **57) Find g(f(-1))** g(f(-1)) = 2 **58) Find f(g(f(o)))** f(g(f(0))) = 0 **59) Find functions f and g such that h(x) = f(g(x)). h(x) = (2x - 5)^3** * f(x) = x^3 * g(x) = 2x - 5 **60) Find the inverse. Is the inverse a function? y = 5 - x^2** y = ±√(5 - x) * Not a function **61) Find f^-¹ . f(x) = √(2x - 3)** f^-¹(x) = (x^2 + 3)/2 **62) Is the following function one-to-one? f(x) = 3x^3 + 1** Yes **68.) The graph of k(x) is shown above. Which of the following best describes the graph of k(x)?** Decreasing and concave up **69. What is the average rate of change of g over the interval 0 ≤ x ≤ 6?** 3/5 **70. For the function f(x) = √8 - 2x find the average rate of change over the interval [−4, 2].** -1/3 **71. The function k(x) = ax^n satisfies the end-behavior below. Which of the following must be true about the function k?** n is odd, and a > 0 **72. The polynomial function g(x) is graphed above. How many points of inflection does the graph of g(x) have?** Three **73. Let f be the function given by f(x) = (1/3)x^4 - (2/3)x^3 - (5/2)x^2 +3x** f has a global minimum of approximately -7.190 **74. The graph of a rational function f is shown at right, along with the horizontal and vertical asymptotes of f** **A (i) Write the equation of the line that is the vertical asymptote of f.** x = -2 **(ii) Write the equation of the line that is the horizontal asymptote of f.** y = 2 **B (i) As x increases without bound, use limit notation to express the end behavior of the function.** lim(x→∞) f(x) = 2 **(ii) Evaluate the following limits:** * lim(x→-∞) f(x) = -∞ * lim(x→2-) f(x) = -∞ * lim(x→2+) f(x) = ∞ **C (i) Find all the intervals for which f(x) ≥ 0.** (-∞, -4] U (-2, 2) U (2, ∞) **(ii) Write a possible equation, in factored form, for the rational function f(x).** f(x) = 2(x + 4)(x - 2) / (x + 2)(x - 2) **75. Which of the following is the equation of the slant asymptote for the graph of g? g(x) = (6x^2 - 3x + 15) / (x + 2)** y = 6x - 15 **76. Write the nth term of the geometric sequence as a function of n. 5/2, 15/8, 45/32, 135/128…** a_n = 5/2(3/4)^(n - 1) **77. A gardener found an invasive flying species that looks similar to a bee in the garden. This wasp-like insect can become a problem for homeowners and gardeners. They can overtake the native bumblebee population.** **You have called an insect specialist to help you keep the good bees and spray insecticide on the wasp-like species that is ruining the garden. The population for each insect is given by the equations below, where B(t) is the population of bumblebees, and t is in days. W(t) is the population of the wasp-like bees, t days after being sprayed.** * **Bumblebee species: B(t) = 1200e^(0.02t)** * **Wasp-like species: W(t) = 2500(1/2)^(0.07t)** **A. What is the initial population of each species when the specialist arrives at t = 0?** * B(0) = 1200 bumblebees * W(0) = 2500 wasps **B. How many days will it take for the bumblebee species to reach a population of 3000 bees?** t ≈ 46 days **C. How long will it take for the wasp-like species to be cut in half after the population has been sprayed?** t ≈ 25 days **The end of the document.**