How many points is each AP Precalculus free-response question worth?

Each of the four questions is worth 6 points. Together they make up 37.5% of your score.

Which free-response questions allow a calculator?

FRQ 1 and FRQ 2, in Section II Part A, require a calculator. FRQ 3 and FRQ 4, in Part B, are done without one.

How many decimal places should my answers have?

Give decimal answers correct to three places after the decimal point, unless the question says otherwise. Keep full precision until the last step.

Do I need to show work if I used Desmos?

Write the equation or expression you used, then the answer. Graders can only award points for what's on the page, so a setup in math notation is good practice.

How to earn points on AP Precalculus free response

Updated November 10, 2026

Free response is 37.5% of your AP Precalculus score. There are four questions, and each one is a set type in a set position. Knowing the four types ahead of time means you won't spend exam minutes working out what a question wants.

This guide covers the habits that matter on every question, then goes through the four types one at a time: what each asks, what earns points, and where points usually slip away.

The setup of the section

Four questions, 6 points each. Each question is split into parts, usually (A), (B), and (C), and each part often has a (i) and a (ii).
Part A has a calculator. FRQ 1, Function Concepts, and FRQ 2, Modeling a Non-Periodic Context, share 35 minutes. Desmos is built into Bluebook, and an approved handheld graphing calculator is allowed too.
Part B has no calculator. FRQ 3, Modeling a Periodic Context, and FRQ 4, Symbolic Manipulations, share another 35 minutes.
Part A and Part B are each 18.75% of your score. That's 35 minutes for two questions, or about 17 minutes per question.
Prompts are on screen, and your answers are handwritten. The full layout of the exam is in the exam format guide.

Habits that apply to every question

Show the setup, then the answer. Write the equation you solved or the expression you evaluated before the number. Graders can only award points for work they can see, and a bare decimal gives them nothing to read. See how to use Desmos on the exam.
Use math notation, not calculator steps. Write $f(x) = 3$ , not "intersect button."
Give decimals to three places unless the question says otherwise, and don't round until the end.
Answer in the context. If the question is about people, years, or feet, your final sentence should say people, years, or feet.
Give a reason when one is asked for. "Give a reason for your answer" means a sentence that names the function and says what it does.
Use your earlier answer in a later part, even if you aren't sure it's right, and show how you used it.
Keep going if you get stuck. Later parts often stand on their own.

FRQ 1: Function Concepts

What it asks. You get two functions, often one as a formula and one as a table or graph. The parts ask you to evaluate compositions, find an inverse, solve an equation with the calculator, and describe behavior: where a function is increasing or decreasing, its end behavior, or its asymptotes.

What earns points: correct values for each piece of a composition, an inverse written as a function of $x$ , every solution in the stated interval, and limit notation where end behavior is asked for.
Common lost points: evaluating $g(f(2))$ in the wrong order, leaving the inverse as " $x = \ldots$ " in terms of $y$ , giving one solution when there are two, and describing end behavior in words when limit notation was requested.

Composition, inverse, and a calculator solve

Let

f(x) = \frac{6}{x + 1}

for

x > -1

, and let

g(x) = x^2 - 4

. (a) Find

g(f(2))

. (b) Find an expression for the inverse function of

f

. (c) Find the

x

-coordinate of the point where the graphs of

f

and

g

intersect.

(a) Work from the inside out. $f(2) = \frac{6}{3} = 2$ , so $g(f(2)) = g(2) = 4 - 4 = 0$ .

(b) Write $y = \frac{6}{x + 1}$ . Then $x + 1 = \frac{6}{y}$ , so $x = \frac{6}{y} - 1$ . Swap the letters: $f^{-1}(x) = \frac{6}{x} - 1$ .

(c) Write the equation first: $\frac{6}{x + 1} = x^2 - 4$ . Then graph $y = f(x)$ and $y = g(x)$ in Desmos and click where they cross. For $x > -1$ they cross once, at $x \approx 2.401$ .

Answer: (a) $0$ . (b) $f^{-1}(x) = \frac{6}{x} - 1$ . (c) $x \approx 2.401$

FRQ 2: Modeling a Non-Periodic Context

What it asks. A situation, usually with a table of data, that fits a linear, quadratic, exponential, or logarithmic model. You choose or build the model, use it to predict, and explain what a number means or why the model stops working.

What earns points: a model with its parameters stated, a prediction with units, and an interpretation that names the quantity, the input value, and the units.
Common lost points: rounding a growth factor too early, giving a prediction with no units, and saying a model has a "limitation" without saying what goes wrong. For regression questions, see the regression guide.

An exponential model from two values

A town had

4800

residents in 2020 and

5400

residents in 2024. Its population is modeled by

P(t) = ab^t

, where

t

is years since 2020. (a) Find

a

and

b

. (b) Use the model to predict the population in 2030. (c) According to the model, during what year does the population first reach

6000

(a) $P(0) = 4800$ , so $a = 4800$ . Then $4800b^4 = 5400$ , so $b^4 = 1.125$ and $b = 1.125^{1/4} \approx 1.030$ .

(b) 2030 is $t = 10$ . Use the unrounded $b$ : $P(10) = 4800(1.125)^{10/4} \approx 6443.511$ , so about $6444$ residents. With $b = 1.030$ you'd get about $6451$ , which shows why you keep full precision in the calculator.

(c) Write $4800b^t = 6000$ . Graph both sides and click the crossing: $t \approx 7.578$ . That's partway through the eighth year after 2020.

Answer: (a) $a = 4800$ , $b \approx 1.030$ . (b) About $6444$ residents. (c) During 2027

FRQ 3: Modeling a Periodic Context

What it asks. A repeating situation, such as a wheel, a tide, or a daily cycle, given as a description, a table, or a graph. You build a sinusoidal function from the highest value, the lowest value, the period, and the starting point, then use it and describe how the quantity changes. There's no calculator, so the values come out of the unit circle.

What earns points: the amplitude, midline, and period as numbers, a model whose maximum or minimum is at the right time, exact values like $\cos\frac{\pi}{3} = \frac{1}{2}$ shown in the work, and a reason for each statement about increasing, decreasing, or concavity.
Common lost points: using the period where $b$ belongs (it's $b = \frac{2\pi}{\text{period}}$ ), a phase shift with the wrong sign, and saying "increasing" without saying whether the rate of change is increasing or decreasing.

Increasing at a decreasing rate

The height, in meters, of a marker on a turning wheel is

h(t) = 5 + 3\sin\left(\frac{\pi t}{6}\right)

, where

t

is in seconds. On the interval

0 < t < 12

, where is

h

increasing and its graph concave down? Give a reason.

The period is $\frac{2\pi}{\pi/6} = 12$ . Sine starts on the midline going up, so $h$ reaches its maximum of $8$ a quarter period in, at $t = 3$ , and its minimum of $2$ at $t = 9$ .

$h$ is increasing on $(0, 3)$ and $(9, 12)$ . Between the midline and a maximum, a sinusoid is concave down, so the graph is concave down on $(0, 6)$ , where it's above the midline $h = 5$ .

Write: "On $(0, 3)$ , $h$ is increasing because it rises from the midline to its maximum, and it's concave down, so the rate of change of $h$ is decreasing there." The average rates of change over $[0, 1]$ , $[1, 2]$ , and $[2, 3]$ are about $1.5$ , $1.098$ , and $0.402$ meters per second, which back this up.

Answer: $0 < t < 3$ , where $h$ is increasing at a decreasing rate

A full walkthrough of building the model is in the sinusoidal models guide.

FRQ 4: Symbolic Manipulations

What it asks. Algebra by hand with exponential, logarithmic, and trigonometric expressions. You rewrite an expression in a requested form, solve an equation exactly, and sometimes state a domain or reject a value.

What earns points: an exact answer (like $\ln 3$ , not $1.099$ ), the property you used visible in the work, the requested form exactly, and every solution in the stated interval.
Common lost points: dividing by $\sin\theta$ and losing the solutions where $\sin\theta = 0$ , missing the second angle in a full turn, and keeping a value that makes a logarithm's input negative. The exponential and log equations guide covers the rules.

Two equations that hide a quadratic

Solve each equation exactly. (a)

e^{2x} - 5e^x + 6 = 0

. (b)

2\sin^2\theta - \sin\theta - 1 = 0

on the interval

[0, 2\pi)

(a) Let $u = e^x$ , so $e^{2x} = u^2$ . Then $u^2 - 5u + 6 = 0$ factors as $(u - 2)(u - 3) = 0$ . So $e^x = 2$ or $e^x = 3$ , which gives $x = \ln 2$ or $x = \ln 3$ . Both are fine, since $e^x$ can equal any positive number.

(b) Let $u = \sin\theta$ . Then $2u^2 - u - 1 = (2u + 1)(u - 1) = 0$ , so $\sin\theta = 1$ or $\sin\theta = -\frac{1}{2}$ .

$\sin\theta = 1$ only at $\theta = \frac{\pi}{2}$ . $\sin\theta = -\frac{1}{2}$ in the third and fourth quadrants, at $\theta = \frac{7\pi}{6}$ and $\theta = \frac{11\pi}{6}$ .

Answer: (a) $x = \ln 2$ or $x = \ln 3$ . (b) $\theta = \frac{\pi}{2}$ , $\frac{7\pi}{6}$ , $\frac{11\pi}{6}$

Writing reasons that earn the point

Reasoning points go to a sentence that names the function, says what it does, and says where. These patterns cover most of what the four questions ask.

Increasing or decreasing: " $f$ is decreasing on $(2, 5)$ because its values get smaller as $x$ increases there."
How the rate changes: "The average rates of change over consecutive equal intervals are $-3$ , $-2$ , and $-1$ . They're increasing, so the graph of $f$ is concave up."
End behavior: " $\lim_{x \to \infty} f(x) = 2$ , so the graph has a horizontal asymptote at $y = 2$ ."
Choosing a model: "The outputs have a roughly constant ratio over equal input intervals, so an exponential model fits."
Interpreting a value: " $P(10) \approx 6444$ means the model predicts about 6444 residents in 2030."

For practice questions of all four types with rubrics, see the full AP Precalculus course.

Keep going

Practice on exam-style questions

Intersect Prep has 349 multiple-choice questions across all 3 exam units, with a full solution for each one. Create a free account to try a full lesson with the calculator. One payment of $29 unlocks the course for 3 months, or $39 for 6 months.

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Questions

How many points is each AP Precalculus free-response question worth?: Each of the four questions is worth 6 points. Together they make up 37.5% of your score.
Which free-response questions allow a calculator?: FRQ 1 and FRQ 2, in Section II Part A, require a calculator. FRQ 3 and FRQ 4, in Part B, are done without one.
How many decimal places should my answers have?: Give decimal answers correct to three places after the decimal point, unless the question says otherwise. Keep full precision until the last step.
Do I need to show work if I used Desmos?: Write the equation or expression you used, then the answer. Graders can only award points for what's on the page, so a setup in math notation is good practice.