AP Precalculus free-response questions: how every point is scored

The four questions

The four questions are always the same four types, in the same order, and each has parts A, B and C. You get two 35-minute parts: questions 1 and 2 with a graphing calculator, then questions 3 and 4 without one.

Every free-response point counts the same. One point is $\frac{1}{24}$ of 37.5%, about 1.6% of the exam score, and one multiple-choice question is worth about 1.5%. So a small point here, like the sentence that finishes an explanation or the third decimal place on an answer, counts as much as a multiple-choice question.

The questions appear on screen in the Bluebook app, but you write your answers by hand in a paper booklet, and only the booklet is scored. Within a part you can move between its two questions until time runs out. On the 2026 exam the on-screen clock turned red at five minutes, and the proctor gave no time warnings, so keep an eye on it yourself.

Thirty-five minutes for two questions is 17½ minutes each. Students had 15 through 2026, so the 2027 exam gives you a little more time to write explanations.

Where students lose points

Each fall the College Board publishes a Chief Reader report: the faculty member who leads the scoring describes how students did on each question and gives the average score on every point. The reports for 2024 and 2025, the exam’s first two years, show the same pattern both times, and you can plan your studying around it.

Most students earn the points for reading and computing. Reading a value off a graph, writing equations from data, computing an average rate of change, finding the heights on a sinusoid: more than half of students earned each of these, both years.

The points for explaining are rarely earned. Question 2’s two explanation points went to between 4% and 8% of students, and the reason in question 1 to 15% in 2024 and 30% in 2025. The 2024 report noted that many explanations stated a correct general rule without evidence from the function itself, and the 2025 report stressed that specific values were required. The section on writing reasons shows what the guidelines accept.

Question 4 is the hardest question on the exam. It had the lowest mean both years, and its last point went to 4% and then 10% of students. In question 3, the point for $b$ and $c$, the constants that come from the period and the starting position, went to 20% of students in 2024 and 18% in 2025.

Added up, the average student earned 9.2 of the 24 points in 2024 and 10.7 in 2025.

Rules that apply to every question

These rules come from the general notes in the scoring guidelines and from the question 4 directions. Students lose points to them every year even when the math is right.

Give decimals to three places

In questions 1 and 2, a decimal answer has to be correct to three places after the decimal point. Rounding and truncating are both accepted, so 5.4828 can be written 5.483 or 5.482, and trailing zeros can go: 2.25 is fine for 2.250.

A correct answer with too few digits is a “decimal presentation error.” The first one in a question costs that point; later decimal-presentation errors in the same question do not automatically lose another point if the response is otherwise complete and correct. An inaccurate calculation from early rounding is a different error. The 2026 guidelines, and the 2025 guidelines for question 2, also count an exact answer where a decimal was asked for, such as $\frac{36}{\ln 92}$ for 7.961, as a decimal presentation error.

Show the work when the question asks for it

When a part says “show the work that leads to your answer” or “show the computations,” the answer alone doesn’t earn the point. In 2025, question 1 asked for $h(1) = g(f(1))$, and students who wrote only 0.333 lost the point: the readers needed to see $f(1) = 1.75$ or $g(1.75)$ on the way. The question 4 directions ask for work in every part, and most of its points require it. In parts without that instruction, a correct answer stands on its own.

A wrong answer next to a right one cancels it

List a correct answer beside an incorrect one and you usually lose the point. The guidelines say it in several places: a list of answers earns the point only with “no incorrect values” in it, and an extra limit statement “must be correct.” In 2025 some students answered $f^{-1}(3.5)$ with both “0” and “no solution.” Commit to one answer.

Say what is changing

Readers are strict about the subject of a sentence. In 2024, “increasing at a decreasing rate” with no subject was read as describing the rate of change of $h$, which was actually decreasing, so it didn’t earn the point. With the subject, “$h$ is increasing at a decreasing rate,” or simply “the rate of change of $h$ is decreasing,” it did. The 2025 and 2026 released versions also asked for concavity, and their points needed both. Also, don’t describe how the rate of change itself is changing. “The rate of change is decreasing at an increasing rate” loses the point even when it’s true, because the guidelines say that conclusion takes calculus.

Which calculator, and when

In the calculator parts you can use the Desmos graphing calculator built into Bluebook or a handheld from the College Board’s approved list, which includes the TI-84 and TI-Nspire families. The website and app versions of Desmos are fine all year but not on exam day. Put a handheld in radian mode.

Calculator work still has to be written down, because the reader sees your booklet, not your screen. Write the equation you solved and the solution, or the regression you ran and the values it gave. And 62.5% of the exam score comes from parts with no calculator at all, so practice the algebra by hand even when a calculator could do it.

Question 1: Function Concepts

Question 1 gives you two functions in two forms: one as a graph or a table, the other as a formula. The formula was exponential in 2024, a cubic in 2025 and a logarithm in 2026. The parts move from reading values, to calculator work, to a reason in words.

Part C(ii) is the point students miss most on this question. In 2024 the question asked whether $f$ has an inverse, and most students knew it doesn’t. What many of them wrote was a general rule, such as “$f$ isn’t one-to-one,” and the guidelines say a rule isn’t enough. The reason has to point at this graph: $f(-3) = f(0) = f(3) = 1$, so the output 1 comes from three different inputs, and an inverse would have to send 1 back to three places at once, which no function can do. Only 15% of students wrote something like that.

Worked example

We wrote this question to match the real question 1 in layout and difficulty, with our own six-point practice rubric based on the published task model. Try it on paper before you read the response.

Lessons on the ideas question 1 keeps testing: composition, inverses, end behavior and asymptotes, and how outputs change for each function type.

• Topic 2.7 · 22 minComposition of Functions
• Topic 2.8 · 19 minInverse Functions
• Topic 1.6 · 19 minPolynomial Functions and End Behavior
• Topic 1.7 · 23 minRational Functions and End Behavior
• Topic 1.9 · 22 minRational Functions and Vertical Asymptotes
• Topic 1.3 · 25 minRates of Change in Linear and Quadratic Functions
• Topic 2.2 · 21 minChange in Linear and Exponential Functions

Question 2: Modeling a Non-Periodic Context

Question 2 is a story with numbers: daily sales of a video game in 2024, plays of a song in 2025, a car losing value in 2026. You’re given a model with two or three unknown constants, and you build it from the data, use it, and say where it stops working. It’s the second-lowest-scoring question, and the explanation points in its last two parts are almost never earned. For 2027 the College Board changed its layout; the 2024–2026 version comes first because the released questions use it, and the new layout follows.

The parts, 2024 to 2026

The 2025 Chief Reader named a mistake in B(ii) worth knowing: students used a line with the right slope through the wrong point. The song data had three points, and they started from the middle one, $(2, 30)$, which isn’t on the secant line between $t = 0$ and $t = 4$. The estimate has to start from one of the two endpoints of the interval you used for the rate.

The secant-line explanation in B(iii)

Part B(iii) asked the same thing all three years: explain why every estimate made with the average rate of change is below (or above) the model between the two data points. It’s gone from the 2027 sample, but the idea is still part of the course.

The average rate of change between two points is the slope of the secant line through them, so every estimate from B(ii) is a point on that line. Whether the line sits below or above the graph depends only on concavity. A concave-down graph bulges above any secant line between two of its points, so every estimate on the secant line is below the graph. A concave-up graph sags below its secant lines, so the estimates are too high. That was 2026’s twist: the car’s value was concave up, and the estimates were too high.

The 2024 report’s diagnosis was that students mixed up the function with its rate of change. Keep them apart: “$D$ is increasing” is about the function, “concave down” is about its rate of change, and only the concavity decides which side of the graph the secant line is on.

Part C: where the model stops making sense

Part C asks about the model’s limits in the story. The point goes to answers that put two facts side by side: what the model does, and what the real quantity can or can’t do.

• 2024, video game sales. Daily sales fell after day 91, but the model $G$ keeps increasing. The two matched on day 91, so the gap between them grows every day after. Earned by 8% of students.
• 2025, song plays. A total number of plays can never go down. The quadratic model peaks at $t = 11$ and decreases after that, so it only works up to $t = 11$. Earned by 5%.
• 2026, a car’s value. The owner will donate the car when its value reaches 2 thousand dollars, and at that moment it loses all its value. Solving $V(k) = 2$ gives $k \approx 22.444$, so the model applies on $0 \le t \le 22.444$.

Each of these answers puts two facts together: what the model does, and what the real quantity can or can’t do. The 2025 report noted that few responses did both: $D$ decreases after $t = 11$, while the total number of plays never decreases.

What changes in 2027

The College Board restructured question 2 for May 2027. In its updated sample question, part B(iii) is gone. Part B is the average rate of change and what it means in context. Part C is worth two points: write and solve an equation for when the model reaches a threshold, then explain the domain restriction that the answer gives. The secant-line reasoning is still part of the course, so keep it; the example below is in the new layout.

Worked example in the 2027 layout

Our own question and six-point practice rubric, built around the College Board’s updated sample tasks. Use a graphing calculator.

Lessons on building a model from data, average rates of change, and the assumptions behind a model.

• Topic 1.3 · 25 minRates of Change in Linear and Quadratic Functions
• Topic 1.13 · 19 minFunction Model Selection and Assumption Articulation
• Topic 1.14 · 23 minFunction Model Construction and Application
• Topic 2.5 · 23 minExponential Function Context and Data Modeling
• Topic 2.6 · 18 minCompeting Function Model Validation
• Topic 2.14 · 20 minLogarithmic Function Context and Data Modeling

Question 3: Modeling a Periodic Context

Question 3 describes something that goes around or back and forth: a point on a rolling tire in 2024, a vibrating guitar string in 2025, a waterwheel in 2026. There’s no calculator. You turn the story into a sinusoid in three steps, and the first two steps carry 4 of the 6 points.

The graph has been the same picture on all three exams so far. It shows two full cycles and a dashed midline, with no axes and no scale, and five labeled points in the same places. F is a maximum, G is on the midline going down, J is a minimum, K is on the midline going up, and P is the next maximum. Each point is a quarter of a period after the one before it.

Step 1: the five points (part A)

1. Read the story for the highest value, the lowest value and the period. The midline is halfway between the highest and lowest values.
2. The heights go maximum, midline, minimum, midline, maximum. For the 2025 guitar string that’s 2, 0, −2, 0, 2 millimeters. This is the first point, and about three students in four earned it.
3. Find one time when the point is at its maximum. That’s F. Add a quarter period for each point after it. In 2025 the string starts at its highest position at $t = 0$ and vibrates 200 times a second, so the period is $\frac{1}{200}$ second and the five times are $0$, $\frac{1}{800}$, $\frac{2}{800}$, $\frac{3}{800}$ and $\frac{4}{800}$.

The times are where students lose points: 39% earned it in 2024 and 34% in 2025. The 2025 report reminds teachers that the period is the reciprocal of the frequency. “200 times in 1 second” is a frequency; one cycle takes $\frac{1}{200}$ of a second.

Step 2: $a$, $b$, $c$ and $d$ (part B)

Every year so far the form has been $h(t) = a\sin(b(t + c)) + d$.

• $d$ is the midline, and $a$ is the amplitude, half the distance from the maximum to the minimum. Take $a$ positive.
• $b = \frac{2\pi}{\text{period}}$. A period of 2 seconds gives $b = \pi$; a period of $\frac{1}{200}$ second gives $b = 400\pi$.
• $c$ comes from the first maximum. With $a > 0$, the sine curve peaks when $b(t + c) = \frac{\pi}{2}$, which happens when $t + c$ is a quarter of a period. So if F is at time $t_F$, then $c = \frac{\text{period}}{4} - t_F$.

Check it on the 2024 tire. The period is 2 seconds, so $b = \pi$, and the first peak on the graph is at $t = \frac{3}{2}$. Then $c = \frac{2}{4} - \frac{3}{2} = -1$ and $h(t) = 9\sin(\pi(t - 1)) + 9$. At $t = \frac{3}{2}$ that gives $9\sin\frac{\pi}{2} + 9 = 18$, the top of the tire.

A negative $a$ is allowed too, with a different $c$. And if your part A times are shifted but still a quarter period apart, the guidelines let you earn the part B points with constants that match your own points. Consistency is rewarded, so finish part B even if you doubt part A.

Step 3: behavior and concavity (part C)

Part C picks two neighboring labeled points and asks about the function between them. C(i) is multiple choice: positive or negative, increasing or decreasing. Positive and negative mean above or below zero, not above or below the midline. On the 2024 tire the height was positive except at the instants W touched the ground.

C(ii) asks how the rate of change is changing, and sinusoids have a shortcut. Above the midline the graph is concave down; below it, the graph is concave up. Concave down means the rate of change is decreasing, and concave up means it’s increasing. In 2026 the interval ran from J, the minimum, to K, the midline going up. That stretch is below the midline, so the answer was “concave up, so the rate of change of $h$ is increasing,” even though $h$ itself was negative there.

Worked example

Our own question and six-point practice rubric, in the same layout and without a calculator. Its part A graph is the standard one described above: two cycles with a dashed midline, F at a maximum, G on the midline going down, J at a minimum, K on the midline going up and P at the next maximum.

Lessons on concavity and rates of change, and on building sinusoidal models from a story.

• Topic 1.1 · 20 minChange in Tandem
• Topic 3.1 · 18 minPeriodic Phenomena
• Topic 3.5 · 16 minSinusoidal Functions
• Topic 3.6 · 20 minSinusoidal Function Transformations
• Topic 3.7 · 18 minSinusoidal Function Context and Data Modeling

Question 4: Symbolic Manipulations

Question 4 is algebra with exponentials, logarithms and trig functions, with no calculator and no real-world context. It has the lowest average of the four questions, 1.28 out of 6 in 2024 and 1.93 in 2025. The 2024 Chief Reader wrote that students across the exam showed more success with concepts than with algebraic manipulation, and pointed to this question’s scores as the evidence.

The layout hasn’t changed: part A solves two equations, part B rewrites two expressions in a required form, and part C is one harder equation, worth two points.

In 2025, $x = 3^2$ instead of 9, and $x = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$ instead of $\frac{\pi}{6}$, didn’t earn the points, because neither was evaluated. Rationalizing denominators isn’t required, though: $\frac{1}{\sqrt{2}}$ is fine.

The moves that keep coming back

Part C needs every solution

Part C’s last point went to 4% of students in 2024 and 10% in 2025. The reports and the 2026 guidelines point to these habits.

• Undo the outside function first. In 2024, $\cos^{-1}(\tan(2x)) = 0$ means $\tan(2x) = \cos 0 = 1$; the report named this step as a gap.
• Look for a quadratic. In 2025, $e^{2x} - e^x - 12 = 0$ is a quadratic in $e^x$, which few responses recognized.
• Solve for the inside angle first. For $\tan(2x) = 1$, solve for $2x$: $2x = \frac{\pi}{4} + n\pi$. Only then divide by 2, and divide the $n\pi$ as well.
• Look for every value of the trig function. $\tan^2(3x) = 1$ means $\tan(3x) = 1$ or $\tan(3x) = -1$. In 2026 one of the three answers came from the $-1$.
• Check the domain and the interval. $e^x = -3$ has no solution. When the question gives an interval, such as $\left[0, \frac{\pi}{2}\right]$ in 2026, the guidelines want the actual values in it, not only a general formula.

Worked example

Our own question and six-point practice rubric. Give exact real answers, finish the requested forms, and show your work without a calculator.

Lessons on the algebra question 4 tests. For the identities and exact values you need from memory, see the AP Precalculus formula sheet.

• Topic 2.12 · 21 minLogarithmic Function Manipulation
• Topic 2.13 · 20 minExponential and Logarithmic Equations and Inequalities
• Topic 3.9 · 18 minInverse Trigonometric Functions
• Topic 3.10 · 19 minTrigonometric Equations and Inequalities
• Topic 3.12 · 21 minEquivalent Representations of Trigonometric Functions

Writing reasons that earn the point

The points for explaining in words are among the least earned on the exam: question 1 C(ii) (15% and 30% in 2024 and 2025), question 2’s last parts (4% to 8%) and question 3 C(ii) (38% and 51%). The scoring guidelines reward the same pattern each time, so it can be learned.

1. Name the property in the course’s words: concave down, increasing on its whole domain, proportional over equal-length input intervals, the secant line.
2. Point at this function. Give the specific values or the interval: $f(-3) = f(0) = f(3) = 1$, or “on $0 < t < 4$,” or $\frac{7}{14} = \frac{3.5}{7} = 0.5$.
3. Finish the thought. End with the claim you were asked to support, joined by “so” or “because.”

The reports add two pieces of advice. Name things instead of writing “it”: the 2024 report suggests teachers show students vague responses, full of “it,” next to precise ones. And reuse the course’s vocabulary (input and output values, rate of change, average rate of change, concave up and down, secant line) because the guidelines are written in it.

What the instruction words ask for

What changes in May 2027

The course content is the same. The College Board changed the multiple-choice counts, timing and question 2, and made a minor wording change to sample question 1 C(i) without changing its scoring.

• Each free-response part is 35 minutes instead of 30, so you have about 17½ minutes a question instead of 15.
• Question 2 is restructured. In the updated sample, the secant-line explanation, B(iii), is gone, B(ii) asks what the average rate of change means in context, and part C is worth two points. The details are in the question 2 section.
• Multiple choice changed too. Part A is 29 questions in 65 minutes (it was 28 in 80), and Part B is 13 questions in 40 minutes (it was 12).

The released questions from 2024 to 2026 are still the closest practice to the real exam for questions 1, 3 and 4, and for most of question 2. For question 2’s new parts, use the College Board’s updated sample in the Course and Exam Description and the practice question above. For the rest of the exam, see the 2027 exam format.

Practice with the released questions

All twelve released questions are free on AP Central with their scoring guidelines, and our solutions go through every point. Do them on paper with a 35-minute timer for each pair, then score yourself with the official guidelines, strictly. The guidelines show which wording earns a point; be strictest with yourself on the explanation points.

Try the 2026 questions first, mark them with the official guidelines, and record each question’s score below. The 2025 and 2024 solution pages have the same score tracker for their questions.

Turn your points into an estimated AP score with the score calculator. For the formulas, identities and exact values to know by heart, use the formula sheet and the AP Precalculus cheat sheets.