Learn Extracted exam questions AP Precalculus 2026 Free Response
2026 Free Response
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The figure shows the graph of the increasing function $f$ on its domain of $-3 \le x \le 3$.
[Graph of $f$: a smooth increasing S-shaped curve on a coordinate grid, $x$-axis from $-3$ to $3$, $y$-axis from $-2$ to $9$ (gridlines at every integer). Labeled points on the curve: $(-3,-2)$, $(-1,1)$, $(0,1.8)$, $(1,3)$, $(2,5)$, $(3,9)$. The curve rises slowly near the middle (around $x=0$) and rises steeply toward both ends.]
The function $g$ is given by $g(x) = -4.792 + \ln(6x - 6)$.
The function $h$ is defined by $h(x) = (g \circ f)(x) = g(f(x))$. Find the value of $h(2)$ as a decimal approximation, or indicate that it is not defined. Show the work that leads to your answer.
Find all values of $x$ for which $f(x) = 3$, or indicate that there are no such values.
Find all values of $x$, as decimal approximations, for which $g(x) = -1.5$, or indicate that there are no such values.
Determine the behavior of $g$ as $x$-values decrease and get arbitrarily close to $1$. Express your answer using the mathematical notation of a limit.
Is the function $f$ invertible?
Give a reason for your answer in part C (i) based on properties of the function $f$. Refer to points on the graph of $f$ in your reasoning.
A person purchased a car at the end of the year 2019 ($t = 0$). The car's value decreases over time. At the end of the year 2020 ($t = 1$), the car was valued at 27.2 thousand dollars. At the end of the year 2025 ($t = 6$), the car was valued at 14.8 thousand dollars.
The value of the car can be modeled by the function $V$ given by $V(t) = ab^t$, where $V(t)$ is the value of the car, in thousands of dollars, at time $t$, and $t$ is the number of years since the end of 2019.
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 $V(t)$.
Find the values for $a$ and $b$ as decimal approximations.
Use the given data to find the average rate of change of the value of the car, in thousands of dollars per year, from $t = 1$ to $t = 6$ years. 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 value of the car, in thousands of dollars, at $t = 3$ years. Show the work that leads to your answer.
The average rate of change found in part B (i) can be used to determine the secant line for the graph of $V$ on the interval $1 \le t \le 6$. Let $A(t)$ represent the estimate of the value of the car, in thousands of dollars, at time $t$ years, using the secant line. For $A(3)$, found in part B (ii), it can be shown that $A(3) > V(3)$.
In general, $A(t) > V(t)$ for all $t$, where $1 < t < 6$. Use the secant line and the graph of $V$ to explain why this is true.
When the value of the car reaches 2 thousand dollars, the car's owner plans to donate it to an auto mechanic school. As a result, the car will immediately lose all of its value. Explain how this information can be used to determine a domain limitation for the model $V$.
The figure shows a circular waterwheel that rotates in a counterclockwise direction at a constant rate and completes 1 revolution in 10 seconds. At time $t = 0$, point $W$ on the edge of the waterwheel is at a height of 6 feet directly above the center of the waterwheel. The height of point $W$ from the horizontal line through the center of the waterwheel periodically decreases and increases as the waterwheel moves.
[Diagram of a circular waterwheel partially submerged in water. The wheel has spokes radiating from a labeled "Center" point, and hatched paddles around the rim. Point $W$ is labeled at the top of the wheel. An arrow labeled "Counterclockwise" indicates the direction of rotation. A vertical double-headed arrow labeled "Height of $W$" shows the vertical distance from the horizontal dashed line through the center up to $W$. The bottom portion of the wheel is submerged in wavy water. Note: Figure not drawn to scale.]
The sinusoidal function $h$ models the height of point $W$ from the horizontal line through the center of the waterwheel, in feet, as a function of time $t$, in seconds. A positive value of $h(t)$ indicates $W$ is above the line through the center; a negative value of $h(t)$ indicates $W$ is below the line through the center.
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.
[Graph of $h$: a sinusoidal (cosine-shaped) curve with a dashed horizontal midline, showing two full cycles. No axes or scale are shown — only the curve and the midline. Point $F$ is labeled at the first (leftmost) maximum of the curve. Point $G$ is labeled where the curve crosses the midline, descending, between the first maximum and the first minimum. Point $J$ is labeled at the minimum of the curve (between the two maxima). Point $K$ is labeled where the curve crosses the midline, ascending, between the minimum and the second maximum. Point $P$ is labeled at the second maximum of the curve.]
Determine possible coordinates $(t, h(t))$ for the five points: $F, G, J, K,$ and $P$.
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 $J$ is $t_1$, and the $t$-coordinate of $K$ 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 $g(x) = \dfrac{1}{e^6}$ for values of $x$ in the domain of $g$.
Solve $h(x) = 3$ for values of $x$ in the domain of $h$.
The functions $j$ and $k$ are given by
Rewrite $j(x)$ as an expression of the form $7^{(\text{expression})}$.
Rewrite $k(x)$ as an expression in which $\sin x$ appears exactly once and no other trigonometric functions are involved.
The function $m$ is given by $m(x) = \tan^2(3x)$.
Find all input values for $m$ in the interval $\left[0, \dfrac{\pi}{2}\right]$ that yield an output value of $1$.