Learn Extracted exam questions AP Precalculus 2025 Free Response
2025 Free Response
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The function $f$ is decreasing and is defined for all real numbers. The table gives values for $f(x)$ at selected values of $x$.
| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|---|
| $f(x)$ | $14$ | $7$ | $3.5$ | $1.75$ | $0.875$ |
The function $g$ is given by $g(x) = -0.167x^3 + x^2 - 1.834$.
The function $h$ is defined by $h(x) = (g \circ f)(x) = g(f(x))$. Find the value of $h(1)$ as a decimal approximation, or indicate that it is not defined. Show the work that leads to your answer.
Find the value of $f^{-1}(3.5)$, or indicate that it is not defined.
Find all values of $x$, as decimal approximations, for which $g(x) = 0$, or indicate that there are no such values.
Determine the end behavior of $g$ as $x$ increases without bound. Express your answer using the mathematical notation of a limit.
Based on the table, which of the following function types best models function $f$: linear, quadratic, exponential, or logarithmic?
Give a reason for your answer in part C (i) based on the relationship between the change in the output values of $f$ and the change in the input values of $f$. Refer to the values in the table in your reasoning.
A musician released a new song on a streaming service. A streaming service is an online entertainment source that allows users to play music on their computers and mobile devices.
Several months later, the musician began using an app (at time $t = 0$) that counts the total number of plays for the song since its release. A "play" is a single stream of the song on the streaming service. The table gives the total number of plays, in thousands, for selected times $t$ months after the musician began using the app. At $t = 0$, the total number of plays was 25 thousand. At $t = 2$, the total number of plays was 30 thousand. At $t = 4$, the total number of plays was 34 thousand.
| Months after the musician began using the app | $0$ | $2$ | $4$ |
|---|---|---|---|
| Total number of plays for the song since its release (thousands) | $25$ | $30$ | $34$ |
The total number of plays, in thousands, for the song since its release can be modeled by the function $D$ given by $D(t) = at^2 + bt + c$, where $D(t)$ is the total number of plays, in thousands, for the song since its release, and $t$ is the number of months after the musician began using the app.
Use the given data to write three equations that can be used to find the values for constants $a$, $b$, and $c$ in the expression for $D(t)$.
Find the values for $a$, $b$, and $c$ as decimal approximations.
Use the given data to find the average rate of change of the total number of plays for the song, in thousands per month, from $t = 0$ to $t = 4$ months. Express your answer as a decimal approximation. Show the computations that lead to your answer.
Use the average rate of change found in part B (i) to estimate the total number of plays for the song, in thousands, for $t = 1.5$ months. Show the work that leads to your answer.
Let $A_t$ represent the estimate of the total number of plays for the song, in thousands, using the average rate of change found in part B (i). For $A_{1.5}$ found in part B (ii), it can be shown that $A_{1.5} < D(1.5)$.
Explain why, in general, $A_t < D(t)$ for all $t$, where $0 < t < 4$. Your explanation should include a reference to the graph of $D$ and its relationship to $A_t$.
The quadratic function model $D$ has exactly one absolute minimum or one absolute maximum. That minimum or maximum can be used to determine a domain restriction for $D$.
Based on the context of the problem, explain how that minimum or maximum can be used to determine a boundary for the domain of $D$.
For a guitar to make a sound, the strings need to vibrate, or move up and down or back and forth, in a motion that can be modeled by a periodic function.
At time $t = 0$ seconds, point $X$ on one vibrating guitar string starts at its highest position, 2 millimeters above its resting position. Then it passes through its resting position and moves to its lowest position, 2 millimeters below the resting position. Point $X$ then passes through its resting position and returns to 2 millimeters above the resting position. This motion occurs 200 times in 1 second.
The sinusoidal function $h$ models how far point $X$ is from its resting position, in millimeters, as a function of time $t$, in seconds. A positive value of $h(t)$ indicates the point is above the resting position; a negative value of $h(t)$ indicates the point is below the resting position.
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 the sinusoidal function $h$ over two full cycles, with a dashed horizontal midline and two solid horizontal lines marking the maximum and minimum levels; no numerical axes or scale shown. The curve starts at a maximum, labeled point $F$, descends crossing the midline at point $G$, continues down to a minimum labeled point $J$, rises back up crossing the midline at point $K$, and reaches the next maximum labeled point $P$, then the curve continues descending again through the end of the second cycle. Points in left-to-right order: $F$ (first maximum), $G$ (midline, descending), $J$ (minimum), $K$ (midline, ascending), $P$ (second maximum).]
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 $d$.
Refer to the graph of $h$ in part A. The $t$-coordinate of $G$ is $t_1$, and the $t$-coordinate of $J$ is $t_2$.
On the interval $(t_1, t_2)$, 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.
On the interval $(t_1, t_2)$, describe the concavity of the graph of $h$ and determine whether the rate of change of $h$ is increasing or decreasing.
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, $\log_2 8$, $\cos\left(\dfrac{\pi}{2}\right)$, 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, $2x+3x$, $5^2 \cdot 5^3$, $\dfrac{x^5}{x^2}$, 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.
The functions $g$ and $h$ are given by
Solve $g(x) = 4$ for values of $x$ in the domain of $g$.
Solve $h(x) = 3$ for values of $x$ in the interval $\left[0, \dfrac{\pi}{2}\right]$.
The functions $j$ and $k$ are given by
Rewrite $j(x)$ as a single logarithm base 2 without negative exponents in any part of the expression. Your result should be of the form $\log_2(\text{expression})$.
Rewrite $k(x)$ as an expression in which $\tan x$ appears exactly once and no other trigonometric functions are involved.
The function $m$ is given by $m(x) = e^{2x} - e^{x} - 12$. Find all input values in the domain of $m$ that yield an output value of 0.