2024 AP Precalculus Free-Response Questions
Advanced PlacementFree ResponseMathematics
4 questions24 pts
4 gradable24 rubric criteria0 dependencies
q1
Question 1
The figure shows the graph of the function f on its domain of −3.5 ≤ x ≤ 3.5. The points (−3, 1), (0, 1), and (3, 1) are on the graph of f. The function g is given by g(x) = 2.916 · (0.7)^x. (A) (i) The function h is defined by h(x) = (g ∘ f)(x) = g(f(x)). Find the value of h(3) as a decimal approximation, or indicate that it is not defined. (ii) Find all values of x for which f(x) = 1, or indicate that there are no such values. (B) (i) Find all values of x, as decimal approximations, for which g(x) = 2, or indicate that there are no such values. (ii) Determine the end behavior of g as x increases without bound. Express your answer using the mathematical notation of a limit. (C) (i) Determine if f has an inverse function. (ii) Give a reason for your answer based on the definition of a function and the graph of y = f(x).
Rubric Criteria (6)
q1_A_iThe response gives h(3)=2.041 as a decimal approximation, correct to three decimal places by rounding or truncating. The first decimal-presentation error in Question 1 makes this point ineligible.
1.0000pts
q1_A_iiThe response identifies exactly x=-3, x=0, and x=3 as all values for which f(x)=1, with no incorrect values included. Supporting work is not required.
1.0000pts
q1_B_iThe response gives x=1.057 as the solution to g(x)=2, expressed as a decimal approximation correct to three decimal places, with no incorrect values included. The first decimal-presentation error in Question 1 makes this point ineligible.
1.0000pts
q1_B_iiThe response states the end behavior using a limit that includes all four required components: lim, x→∞, the function g, and 0; equivalently, lim_{x→∞}g(x)=0. If an additional complete limit statement is included, it must also be correct.
1.0000pts
q1_C_iThe response states that f does not have an inverse function on its domain −3.5≤x≤3.5.
1.0000pts
q1_C_iiThe response explains, with an implicit or explicit reference to the definition of a function, that f has output values not mapped from unique input values, and supports this with specific function values such as f(−3)=f(0)=f(3)=1. This criterion cannot be awarded if Part (C)(ii) contains an error.
1.0000pts
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q2
Question 2
2. On the initial day of sales (t = 0) for a new video game, there were 40 thousand units of the game sold that day. Ninety-one days later (t = 91), there were 76 thousand units of the game sold that day. The number of units of the video game sold on a given day can be modeled by the function G given by G(t) = a + b ln(t + 1), where G(t) is the number of units sold, in thousands, on day t since the initial day of sales. (A) (i) Use the given data to write two equations that can be used to find the values for constants a and b in the expression for G(t). (ii) Find the values for a and b as decimal approximations. (B) (i) Use the given data to find the average rate of change of the number of units of the video game sold, in thousands per day, from t = 0 to t = 91 days. Express your answer as a decimal approximation. Show the computations that lead to your answer. (ii) Use the average rate of change found in (i) to estimate the number of units of the video game sold, in thousands, on day t = 50. Show the work that leads to your answer. (iii) Let A_t represent the estimate of the number of units of the video game sold, in thousands, using the average rate of change found in (i). For A_50, found in (ii), it can be shown that A_50 < G(50). Explain why, in general, A_t < G(t) for all t, where 0 < t < 91. (C) The makers of the video game reported that daily sales of the video game decreased each day after t = 91. Explain why the error in the model G increases after t = 91.
Rubric Criteria (6)
q2_A_iThe response gives two equations involving a and b that use the given input-output pairs G(0)=40 and G(91)=76; for example, a+b ln(0+1)=40 and a+b ln(91+1)=76.
1.0000pts
q2_A_iiThe response gives the correct decimal values a=40 and b=7.961, each correct to three decimal places, either as standalone values or in an expression for G(t), such as G(t)=40+7.961 ln(t+1). Values translated consistently into units rather than thousands are also acceptable.
1.0000pts
q2_B_iThe response shows computations using the given data values in the quotient [G(91)−G(0)]/(91−0) and gives the average rate of change as 0.396 or 0.395 thousand units per day, correct to three decimal places by rounding or truncating. Units are not required.
1.0000pts
q2_B_iiThe response shows supporting work using the average rate of change from Part (B)(i) and a secant-line calculation from an endpoint, such as 40+r(50−0) or 76+r(50−91), and estimates G(50) as 59.780 thousand units, or a value in [59.750,59.805] when the rate in Part (B)(i) has no decimal-presentation error. A final value of 59 or 60 thousand is acceptable only when the supporting work gives a correct decimal estimate.
1.0000pts
q2_B_iiiThe response explains that A_t lies on the secant line through (0,G(0)) and (91,G(91)), and that because G is concave down on (0,91), or its rate of change is decreasing, this secant line lies below the graph of G on that interval; therefore A_t<G(t) for all 0<t<91.
1.0000pts
q2_CThe response explains that at t=91 the actual daily sales and G(91) agree, while for t>91 actual daily sales decrease and G(t) increases; therefore the absolute error between actual daily sales and the values predicted by G increases each day after t=91.
1.0000pts
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q3
Question 3
3. The tire of a car has a radius of 9 inches, and a person rolls the tire forward at a constant rate on level ground, as shown in the figure. Point W on the edge of the tire touches the ground at time t = 1/2 second. The tire completes a full rotation, and the next time W touches the ground is at time t = 5/2 seconds. The maximum height of W above the ground is 18 inches. As the tire rolls, the height of W above the ground periodically increases and decreases. The sinusoidal function h models the height of point W above the ground, in inches, as a function of time t, in seconds. Note: Figure not drawn to scale. (A) The graph of h and its dashed midline for two full cycles is shown. Five points, F, G, J, K, and P, are labeled on the graph. No scale is indicated, and no axes are presented. Determine possible coordinates (t, h(t)) for the five points: F, G, J, K, and P. [Graph of h and its dashed midline with points F, G, J, K, and P labeled.] (B) The function h can be written in the form h(t) = a sin(b(t + c)) + d. Find values of constants a, b, c, and d. (C) Refer to the graph of h in part (A). The t-coordinate of K is t₁, and the t-coordinate of P is t₂. (i) On the interval (t₁, t₂), which of the following is true about h? a. h is positive and increasing. b. h is positive and decreasing. c. h is negative and increasing. d. h is negative and decreasing. (ii) Describe how the rate of change of h is changing on the interval (t₁, t₂).
Rubric Criteria (6)
q3_A_h_coordinatesThe response gives the correct h-coordinates for all five labeled points: F and P have height 18, G and K have height 9, and J has height 0. A correct set of coordinates for one full cycle, including the listed vertical values, is acceptable.
1.0000pts
q3_A_t_coordinatesThe response gives valid t-coordinates for all five labeled points, with the points occurring at t-values −1/2+2k, 0+2k, 1/2+2k, 1+2k, and 3/2+2k for one specific integer k, paired with the labeled points as shown. Negative t-coordinates and any correct one-cycle shift are acceptable.
1.0000pts
q3_B_vertical_transformationsFor h(t)=a sin(b(t+c))+d, the response gives the correct vertical-transformation values a=9 and d=9, or the equivalent reflected form a=−9 and d=9.
1.0000pts
q3_B_horizontal_transformationsFor h(t)=a sin(b(t+c))+d, the response gives b=2π/2=π and a valid corresponding phase shift: c=−1+2k for a=9, or c=0+2k for a=−9, where k is any integer. Equivalent correct forms based on horizontal shifts and reflections are acceptable.
1.0000pts
q3_C_iThe response identifies choice a, or states that h is positive and increasing, on the interval (t1,t2).
1.0000pts
q3_C_iiThe response states that the rate of change of h is decreasing on (t1,t2), or that h is increasing at a decreasing rate. If concavity is referenced, the graph must correctly be identified as concave down; stating only concave down is insufficient.
1.0000pts
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q4
Question 4
4. Directions: • Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. Angle measures for trigonometric functions are assumed to be in radians. • Solutions to equations must be real numbers. Determine the exact value of any expression that can be obtained without a calculator. For example, log₂ 8, cos(π/2), and sin⁻¹(1) can be evaluated without a calculator. • Unless otherwise specified, combine terms using algebraic methods and rules for exponents and logarithms, where applicable. For example, 2x + 3x, 5² · 5³, x⁵/x², and ln 3 + ln 5 should be rewritten in equivalent forms. • For each part of the question, show the work that leads to your answers. (A) The functions g and h are given by g(x) = e^(x+3) h(x) = arcsin(x/2). (i) Solve g(x) = 10 for values of x in the domain of g. (ii) Solve h(x) = π/4 for values of x in the domain of h. (B) The functions j and k are given by j(x) = log₁₀(8x⁵) + log₁₀(2x²) − 9 log₁₀ x k(x) = ((1 − sin² x)/sin x) sec x. (i) Rewrite j(x) as a single logarithm base 10 without negative exponents in any part of the expression. Your result should be of the form log₁₀(expression). (ii) Rewrite k(x) as a single term involving tan x. (C) The function m is given by m(x) = cos⁻¹(tan(2x)). Find all values in the domain of m that yield an output value of 0. Write your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part.
Rubric Criteria (6)
q4_A_iThe response solves e^(x+3)=10 and gives x=−3+ln(10), or an equivalent correctly evaluated logarithmic expression, with supporting work.
1.0000pts
q4_A_iiThe response solves arcsin(x/2)=π/4 and gives x=2, with supporting work that includes x/2=sin(π/4), x/2=√2/2, or x/2=1/√2.
1.0000pts
q4_B_iThe response rewrites j(x) as a single base-10 logarithm with no negative exponents, obtaining log_10(16/x^2), or an explicitly accepted equivalent such as log_10((4/x)^2), with supporting work. Use of log without the base is acceptable, and domain restrictions need not be stated.
1.0000pts
q4_B_iiThe response rewrites k(x)=((1−sin^2 x)/sin x)sec x as the single term 1/tan x, or an equivalent expression such as cos x/sin x or cot x, with supporting work. Domain restrictions need not be stated.
1.0000pts
q4_C_one_valueThe response provides at least one correct real value of x in the domain of m for which m(x)=0, with supporting work showing the equation reduces to tan(2x)=1; for example, x=π/8.
1.0000pts
q4_C_all_valuesThe response gives all real solutions with no incorrect values, x=π/8+πn/2 where n is any integer, or an equivalent complete form such as x=π/8+πn and x=5π/8+πn, with supporting work.
1.0000pts
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Attribution
This material is the intellectual property of College Board, is not owned by GradeThis, and is shown here for ease of use and demonstration purposes.