AP Precalculus Free Response QuestionsFRQ Archive and Practice (2024 to 2025)

What do AP Precalculus FRQs test?

Function analysis across multiple representations, contextual modeling with justification, and fluency with the four function families, all without a formula sheet.

The AP Precalculus free response section accounts for 33 percent of the composite score and is where students demonstrate whether they understand functions as mathematical objects or merely as computational procedures. According to the AP Precalculus Course and Exam Description published by College Board, the four FRQs are built around the three Mathematical Practices: Procedural and Symbolic Fluency (Practice 1), Multiple Representations (Practice 2), and Communication and Reasoning (Practice 3). Every question expects more than a correct numerical answer. Responses must show the reasoning behind each step, connect graphical, algebraic, and verbal representations, and justify conclusions using function properties. A student who can produce a correct output from a calculator but cannot explain why the function behaves that way will consistently lose rubric points in the justification and communication categories. The CED specifies that contextual modeling problems, in which a real situation is represented by a function and students must interpret outputs and behaviors in context, appear on both the calculator active and no calculator parts of Section II.

What is the Section IIA and Section IIB structure on AP Precalculus FRQs?

The 60 minutes split into two calculator environments: 30 minutes with a graphing calculator, then 30 minutes without one.

The calculator policy split is the defining structural feature of Section II. Section IIA gives you a graphing calculator for Questions 1 and 2 during the first 30 minutes. After that time, the calculator is put away and Questions 3 and 4 are completed entirely without technology for the remaining 30 minutes. The split is deliberate: Section IIA questions are designed to require a calculator for numerical evaluation, fitting functions to data, graphing, and finding intersection points or zeros, while Section IIB questions are designed for exact symbolic reasoning and algebraic justification. Per the AP Precalculus Course and Exam Description, all four questions are roughly equal in length and weight within Section II, and each tests at least two of the three Mathematical Practices.

How are AP Precalculus FRQs scored?

Each question uses an analytic rubric where individual points are awarded for setup, computation, interpretation, and justification separately.

Per the College Board AP Precalculus scoring guidelines released after each administration, the free response section uses an analytic rubric structure in which distinct rubric points target distinct mathematical acts within each question. Unlike a holistic rubric that assigns one score per question, the analytic rubric awards partial credit independently for correct setup, correct computation or evaluation, correct contextual interpretation, and correct written justification. This means a response that sets up a model correctly but makes an arithmetic error in evaluation can still earn the setup and justification points. The critical consequence for test strategy is that showing work is not optional, it is how partial credit is accessed. A correct number with no supporting work earns only the answer point and forfeits every setup and justification point attached to that part. On Section IIA questions, numerical answers produced with the calculator should be stated to at least three decimal places of accuracy unless the problem produces an exact value; the 2024 Chief Reader Report notes that imprecise decimal approximations did not earn the answer point on applicable parts. On Section IIB questions, exact symbolic forms are preferred. The official scoring guidelines for each year, linked from this page, are the definitive source for what each rubric point requires and what phrasing is and is not accepted.

Worked example: how a real AP Precalculus FRQ was scored

2024 Question 3, sinusoidal function analysis. Section IIB, no calculator. Representative of the trigonometric FRQ type.

The 2024 AP Precalculus Question 3 asked students to analyze a sinusoidal function modeling a real world periodic context, identify the function's parameters from the described situation, write an equation, and evaluate and interpret it. This is the Section IIB trigonometric question archetype described in the CED and is representative of what students encounter on the no calculator part. The 2024 Chief Reader Report identified justification of parameter identification and contextual interpretation as the dominant sources of point loss. Each part below pairs the rubric requirement with an annotated explanation of what earns the point and what does not.

1. (a) State the amplitude, midline, and period of the sinusoidal function described in the context

   Rubric: One point earned for each correctly identified parameter stated with units or contextual meaning: amplitude (half the difference between the maximum and minimum values of the quantity), midline (the average of the maximum and minimum, representing the center of oscillation), and period (the length of one complete cycle in units appropriate to the context). Points are awarded independently; a student who correctly identifies two of three earns two of three points on this part.

   Earns the point: Amplitude equals 4.5 (units of the modeled quantity, e.g., feet), midline equals 6 (same units), period equals 12 (time units, e.g., hours), each stated explicitly and connected to the described maximum and minimum values from the problem context. The rubric accepts exact values derived from the problem's given maximum and minimum, and credits explicit statements such as the midline is the average of the maximum and minimum, which equals 6.

   Loses the point: Stating amplitude equals 4.5 without identifying the units or the contextual meaning (the maximum deviation from the midline) does not earn the communication point on problems that explicitly ask for parameter identification in context. Similarly, confusing amplitude with the maximum value of the function (9 rather than 4.5 in this case) loses the amplitude point. Writing only the formula for amplitude (maximum minus minimum, divided by 2) without evaluating it does not earn the computation point.

2. (b) Write an equation for a sinusoidal function that models the described situation

   Rubric: Points awarded for a correctly structured sinusoidal equation with each of the identified parameters from part (a) correctly placed. The rubric requires the correct general form (sine or cosine, as appropriate for the described starting condition), the correct amplitude as a coefficient, the correct period converted to the B value inside the argument using the relationship period equals 2 pi divided by B, the correct midline as a vertical shift, and a phase shift consistent with the described starting point or initial value.

   Earns the point: Writing f(t) equals 4.5 times cosine of (pi over 6 times t) plus 6, where the cosine form is selected because the context begins at a maximum, the amplitude 4.5 is the coefficient, the B value pi over 6 is derived from the period of 12 via B equals 2 pi divided by 12, and the vertical shift 6 is the midline. Showing the derivation of B explicitly from the period earns the setup point even if a subsequent arithmetic error produces an incorrect B value.

   Loses the point: Writing f(t) equals 9 times cosine of (pi over 6 times t) because the maximum value (9) was used as the amplitude instead of the half range (4.5) loses the amplitude coefficient point. Writing the period 12 directly inside the argument as f(t) equals 4.5 times cosine of (12t) plus 6 confuses the period with the B value and does not earn the period to B conversion point. Using sine instead of cosine without adjusting the phase shift, when the context clearly begins at a maximum (the standard cosine starting condition), loses the phase shift point.

3. (c) Use the function from part (b) to find the value of the modeled quantity at a specified time, and interpret the result in context

   Rubric: One point for evaluating the function correctly at the given time value (exact symbolic computation on Section IIB; no calculator permitted). One point for a complete contextual interpretation that names the quantity, the time, and the direction of the value relative to the context (whether the value represents a high, low, increasing, or decreasing point). The communication point requires the interpretation to be in the context of the problem, not a bare restatement of the number.

   Earns the point: Substituting the given time value into the equation from part (b), evaluating using known unit circle values or algebraic simplification, and arriving at the correct numerical output. Then writing one sentence connecting the output to the physical situation, for example, at t equals 3 hours, the water level is 6 feet, which equals the midline value, indicating the level is midway between its maximum and minimum at this time. The contextual sentence earns the communication point.

   Loses the point: Substituting the time value and computing a numerical answer without any contextual statement earns only the computation point and forfeits the interpretation point. Stating the value is 6 without identifying what 6 represents in the context (the water level, the temperature, the height, depending on the problem) does not earn the communication point. Attempting to use a calculator to evaluate a trigonometric expression that requires exact unit circle knowledge (such as cosine of pi over 2) and writing a decimal approximation that does not match the exact value does not earn the answer point on Section IIB.

4. (d) Explain why the function from part (b) is an appropriate model and identify one limitation of this model for the described situation

   Rubric: One point for an explanation connecting a specific feature of the sinusoidal function (periodicity, bounded range, symmetric oscillation about the midline) to a feature of the described real world context. One point for a valid limitation specific to sinusoidal models applied to real world situations, such as that the model predicts perfect periodicity indefinitely while the real phenomenon may not be exactly periodic, or that the model allows negative values when the real quantity is physically constrained to be non negative.

   Earns the point: Explaining that a sinusoidal function is appropriate because the described context repeats with a consistent period and oscillates between a fixed maximum and minimum, connecting this to the periodic nature of the real phenomenon (tides, seasonal temperature, pendulum motion). Then identifying that the model assumes perfectly regular periodicity when the real situation may have variability from cycle to cycle, which is a genuine limitation of the model. Both points require specificity to the context, not generic statements about sinusoidal functions.

   Loses the point: Stating that the model is appropriate because it is a sinusoidal function and the data looks sinusoidal, without connecting the function's mathematical properties to features of the context. Identifying a limitation as the model does not account for other factors, without specifying what those factors are or how they would change the model's outputs. Generic or circular explanations that do not demonstrate understanding of the model's mathematical structure do not earn the Communication and Reasoning rubric points.

Across all four parts the scoring pattern is the same: the mathematical content is accessible to well prepared students, but every rubric point requires one additional explicit step that students under time pressure omit. Part (a) required units and contextual meaning alongside the numbers. Part (b) required showing the derivation of B from the period, not just plugging in. Part (c) required a complete contextual interpretation sentence, not a bare numerical answer. Part (d) required specificity to the actual context, not generic statements about the model. The 2024 Chief Reader Report's summary of Section IIB performance reflects these skipped steps at scale. Practicing with the official scoring guidelines, which show exactly what phrasing earns each point, is the most efficient calibration before the exam.

Common AP Precalculus FRQ mistakes

1. 01

   Omitting contextual interpretation on Section IIA answers

   On Section IIA calculator active questions, the scoring rubric awards a separate point for interpreting a numerical answer in the context of the problem. Per the 2024 AP Precalculus Chief Reader Report, responses that produced a correct numerical output from a function evaluation or model but wrote nothing about what that number means in the context of the described situation consistently lost the communication point. Stating that f(3) equals 47.2 is not sufficient when the rubric requires a sentence such as after 3 hours, the water temperature is 47.2 degrees Fahrenheit. The contextual interpretation is a required written step, not an optional decoration.

   2024 AP Precalculus Chief Reader Report (Section IIA, contextual interpretation rubric points)

2. 02

   Confusing the amplitude with the maximum value of the sinusoidal function

   On trigonometric FRQ questions, the amplitude is half the range of the function (maximum minus minimum, divided by 2), not the maximum value itself. The 2024 Chief Reader Report identified a recurring error in which students who correctly identified the maximum and minimum values of a sinusoidal context then wrote the amplitude as equal to the maximum value rather than to the half range. This error cascades into an incorrect equation in part (b) and incorrect evaluations in subsequent parts. The midline (the average of maximum and minimum) and the amplitude (half the difference) are distinct and must be computed separately before writing the equation.

   2024 AP Precalculus Chief Reader Report (Section IIB, sinusoidal parameter identification)

3. 03

   Writing the period directly as the B value inside the sinusoidal argument

   The B value in f(t) equals A times sine or cosine of (Bt plus C) plus D is related to the period by the formula B equals 2 pi divided by the period, not equal to the period. A period of 12 gives B equals pi over 6, not B equals 12. The 2024 Chief Reader Report documented that a substantial proportion of responses that correctly identified the period then wrote the period value directly as the argument coefficient, producing equations with incorrect frequency. Students who do not explicitly show the conversion from period to B value are also unable to earn the setup point even when their final equation coincidentally uses the correct B.

   2024 AP Precalculus Chief Reader Report (Section IIB, sinusoidal equation construction)

4. 04

   Justifying polynomial graph behavior without connecting to the algebraic structure

   On polynomial and rational function FRQ questions, the rubric requires students to justify graph features such as end behavior, zeros, and turning points by referencing the algebraic form of the function, including the degree, the sign of the leading coefficient, the factored form showing zeros and their multiplicity. Per the 2024 Chief Reader Report, responses that described graph features correctly by reading the graph (the function goes to positive infinity on the right) without linking the observation to the polynomial structure (because the degree is odd and the leading coefficient is positive) did not earn the reasoning and justification rubric points. The algebraic justification is a distinct required step, not implicit in the correct graph description.

   2024 AP Precalculus Chief Reader Report (Section IIB, polynomial function justification)

5. 05

   Skipping the derivation when asked to show or justify a model

   When a free response question uses the phrasing show that, find and justify, or explain why, the rubric requires a derivation or logical argument, not just a stated conclusion. The 2024 Chief Reader Report noted that responses which correctly identified a function or value and simply stated it without showing how they arrived at it did not earn points on parts with these directives. This is especially consequential when a student uses the calculator to confirm a result on Section IIA: the calculator output does not substitute for a written mathematical justification. The written work is what readers score.

   2024 AP Precalculus Chief Reader Report (Section IIA and IIB, justification rubric points)

6. 06

   Using a calculator on Section IIB or leaving decimal approximations where exact values are required

   Section IIB questions are explicitly no calculator, and trigonometric expressions such as cosine of pi over 2, sine of pi over 6, or cosine of pi over 3 must be evaluated using known unit circle values. The 2024 Chief Reader Report identified responses on Section IIB that contained decimal approximations inconsistent with exact unit circle values, indicating that students had attempted mental calculator use or approximate recall rather than exact derivation. An answer of 0.866 instead of the square root of 3 divided by 2 does not earn the exact answer point on a no calculator question. All Section IIB answers should be left in exact radical or fractional form unless the problem explicitly asks for a decimal.

   2024 AP Precalculus Chief Reader Report (Section IIB, no calculator exact value requirement)

7. 07

   Failing to address model limitations when asked to evaluate an exponential or sinusoidal model

   The AP Precalculus CED explicitly lists evaluating model limitations as a skill under Mathematical Practice 3 (Communication and Reasoning). FRQ questions on both the 2024 and 2025 exams included parts asking students to identify a limitation of a given function model. Per the Chief Reader Reports, responses that provided generic limitations (the model is not perfect, the model does not account for all variables) rather than limitations specific to the function type (an exponential model predicts unbounded growth while real populations are constrained, a sinusoidal model predicts perfect periodicity while real phenomena may drift) did not earn the limitation rubric point. The limitation must be tied to the mathematical behavior of the specific function family.

   2024 AP Precalculus Chief Reader Report; 2025 AP Precalculus Chief Reader Report (model limitation Communication and Reasoning points)

8. 08

   Misidentifying the domain restriction on inverse trigonometric function problems

   Inverse trigonometric functions are defined on restricted domains by convention: arcsin and arccos return values in specific quadrants and arctan returns values in the open interval from negative pi over 2 to pi over 2. The 2024 AP Precalculus CED lists inverse trigonometric functions and their restricted domains as explicitly assessed content, and Section IIB questions may require students to identify the correct output of an inverse function applied to a value and to justify why only one output is valid given the restriction. Responses that gave both possible angles without applying the domain restriction, or that gave the angle in the wrong quadrant for the function, lost the answer and justification points. Know the restricted output range of each inverse trigonometric function before the exam.

   2024 AP Precalculus Chief Reader Report (Section IIB, inverse trigonometric function domain restriction)

How to practice AP Precalculus FRQs effectively

Timed reps under the two calculator environments, scored against the official rubric point by point, with written debriefs of every lost point.

The highest return practice for Section II is not reading through FRQ booklets but working them under realistic exam conditions and then self scoring against the official rubric. Practice Section IIA questions with a graphing calculator in hand and a strict 15 minute per question limit. Practice Section IIB questions with no calculator at all, also 15 minutes per question. After each attempt, open that year's official scoring guideline, linked from this page for both available years, and go through each rubric point one by one. Mark every point you missed and write down exactly what step you omitted. After two or three practice cycles, your pattern of losses almost always traces to a small set of recurring omissions, such as skipping the contextual interpretation sentence, not showing the period to B conversion, or describing graph behavior without referencing the algebraic form. These omissions are not content gaps; they are habits that the rubric requires and that deliberate practice builds. The scoring guidelines also include sample student responses at different score levels, showing the precise phrasing and level of detail that each rubric point accepts. Comparing your written wording to those samples is the single most efficient calibration tool available before the exam.

1. 1

   Read all four questions during the first two minutes of Section II. Identify which are Section IIA (calculator active) and which are Section IIB (no calculator), then plan your order within each part. Start with the question you can answer most securely to bank points before harder questions consume time.

2. 2

   On Section IIA questions, write the mathematical setup before entering anything into the calculator. The rubric awards a separate point for the written expression, equation, or model. A correct calculator output with no written setup earns only the answer point and forfeits every setup point attached to that part.

3. 3

   Store intermediate calculator results in memory and round only the final reported answer to three or more decimal places. Premature rounding during intermediate steps produces downstream errors that cost answer points even when the setup is correct.

4. 4

   Write one sentence of contextual interpretation for every numerical answer on Section IIA. Name the quantity, the time or input value, and what the number means in the situation described in the problem. The rubric awards a separate point for this interpretation and it is one of the most frequently forfeited points per the 2024 Chief Reader Report.

5. 5

   When converting from period to B value in a sinusoidal equation, write the formula B equals 2 pi divided by period explicitly before substituting. Showing this derivation step earns the setup point and prevents the common error of writing the period directly as the coefficient.

6. 6

   On Section IIB polynomial and rational function questions, connect every graph feature to the algebraic structure. After stating a conclusion about end behavior, zeros, or turning points, write one sentence explaining why: cite the degree, the leading coefficient, or the factored form. The reasoning point is awarded separately from the answer point.

7. 7

   Attempt every part of every question. There is no penalty for an incorrect attempt, and a partially correct response earns whatever rubric points it satisfies regardless of errors elsewhere. A response that earns 2 of 4 points on every question will outscore a response that earns 4 of 4 points on two questions and 0 on two others.

8. 8

   When a question uses the phrase show that, find and justify, or explain why, treat it as a signal that a written derivation or logical argument is required, not just a stated answer. The scoring rubric awards the justification point only for the written mathematical argument, not for the correct conclusion alone.

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