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Precalc

AP Precalculus

Polynomial, rational, exponential, logarithmic, trigonometric, and polar functions, learned as tools for modeling how quantities change together.

Category
Math and computer science
Units
4 units
Exam
Exam May 11, 2027 (in 226 days)
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Unit 1 is free with an account. No card, no trial clock.

What the course covers

A college precalculus course built around one idea: a function describes how two quantities change in tandem. You study four families (polynomial and rational, exponential and logarithmic, trigonometric and polar, and general functions given only as graphs or tables), and for each you learn to read its rate of change, rewrite it in useful equivalent forms, build it as a model from data or a context, and explain its limits. Every unit trains the three AP practices that readers score: procedural and symbolic fluency (most of the exam is taken without a calculator), moving between graphs, tables, formulas, and words, and precise communication, such as limit notation for asymptotes and justifications that cite values from a table. The exam has 42 multiple-choice questions and 4 six-point free-response questions; a graphing calculator is required on about a third of each section, and no formula sheet is given.

4 units, with exam weights

  1. Unit 1

    Polynomial and Rational Functions

    30-40% of MCQ of examFree
    How inputs and outputs change together, average rates of change and what they reveal about concavity, then the polynomial and rational families: zeros and complex zeros, end behavior, asymptotes and holes in limit notation, equivalent forms, transformations, and building polynomial, piecewise, and rational models from contexts and data.
    14 topics
    1. 1.1Change in Tandem
    2. 1.2Rates of Change
    3. 1.3Rates of Change in Linear and Quadratic Functions
    4. 1.4Polynomial Functions and Rates of Change
    5. 1.5Polynomial Functions and Complex Zeros
    6. 1.6Polynomial Functions and End Behavior
    7. 1.7Rational Functions and End Behavior
    8. 1.8Rational Functions and Zeros
    9. 1.9Rational Functions and Vertical Asymptotes
    10. 1.10Rational Functions and Holes
    11. 1.11Equivalent Representations of Polynomial and Rational Expressions
    12. 1.12Transformations of Functions
    13. 1.13Function Model Selection and Assumption Articulation
    14. 1.14Function Model Construction and Application

  2. Unit 2

    Exponential and Logarithmic Functions

    25-40% of MCQ of exam
    Arithmetic and geometric sequences grow into linear and exponential functions: additive versus proportional change, exponential models from ratios and data, residuals and model validation, composition and inverse functions, logarithms as inverses of exponentials, log properties, solving exponential and logarithmic equations, logarithmic models, and semi-log plots.
    15 topics
    1. 2.1Change in Arithmetic and Geometric Sequences
    2. 2.2Change in Linear and Exponential Functions
    3. 2.3Exponential Functions
    4. 2.4Exponential Function Manipulation
    5. 2.5Exponential Function Context and Data Modeling
    6. 2.6Competing Function Model Validation
    7. 2.7Composition of Functions
    8. 2.8Inverse Functions
    9. 2.9Logarithmic Expressions
    10. 2.10Inverses of Exponential Functions
    11. 2.11Logarithmic Functions
    12. 2.12Logarithmic Function Manipulation
    13. 2.13Exponential and Logarithmic Equations and Inequalities
    14. 2.14Logarithmic Function Context and Data Modeling
    15. 2.15Semi-log Plots

  3. Unit 3

    Trigonometric and Polar Functions

    30-35% of MCQ of exam
    Periodic phenomena and the unit circle: sine, cosine, and tangent as functions of radian angle measure, sinusoidal graphs and transformations, sinusoidal models of real contexts, the tangent function, inverse trigonometric functions, trigonometric equations and inequalities, reciprocal functions, identities (Pythagorean, sum and difference, double angle), polar coordinates, polar graphs, and how the distance from the origin changes along a polar curve.
    15 topics
    1. 3.1Periodic Phenomena
    2. 3.2Sine, Cosine, and Tangent
    3. 3.3Sine and Cosine Function Values
    4. 3.4Sine and Cosine Function Graphs
    5. 3.5Sinusoidal Functions
    6. 3.6Sinusoidal Function Transformations
    7. 3.7Sinusoidal Function Context and Data Modeling
    8. 3.8The Tangent Function
    9. 3.9Inverse Trigonometric Functions
    10. 3.10Trigonometric Equations and Inequalities
    11. 3.11The Secant, Cosecant, and Cotangent Functions
    12. 3.12Equivalent Representations of Trigonometric Functions
    13. 3.13Trigonometry and Polar Coordinates
    14. 3.14Polar Function Graphs
    15. 3.15Rates of Change in Polar Functions

  4. Unit 4

    Functions Involving Parameters, Vectors, and Matrices

    Not assessed on the AP Exam of exam
    An optional unit in the official framework, taught where state or local standards require it and never tested on the AP Exam: parametric functions and planar motion, implicitly defined functions and conic sections, parametrizing curves, vectors and vector-valued functions, matrices, determinants and inverses, linear transformations, and matrix models of transitions between two states.
    14 topics
    1. 4.1Parametric Functions
    2. 4.2Parametric Functions Modeling Planar Motion
    3. 4.3Parametric Functions and Rates of Change
    4. 4.4Parametrically Defined Circles and Lines
    5. 4.5Implicitly Defined Functions
    6. 4.6Conic Sections
    7. 4.7Parametrization of Implicitly Defined Functions
    8. 4.8Vectors
    9. 4.9Vector-Valued Functions
    10. 4.10Matrices
    11. 4.11The Inverse and Determinant of a Matrix
    12. 4.12Linear Transformations and Matrices
    13. 4.13Matrices as Functions
    14. 4.14Matrices Modeling Contexts

The exam, part by part

4 parts, 2 h 55 min in all.

  • Section I, Part A: Multiple Choice (no calculator)

    Questions
    29
    Time
    1 h 5 min
    Weight
    43.8%

    No calculator

    Format details

    Four answer choices (A-D), all stand-alone items, taken on screen in Bluebook. Items use graphical, numerical (tables), analytical, and verbal representations, some in real-world contexts and some about modeling. New for May 2027: 29 questions in 65 minutes (was 28 in 80).

  • Section I, Part B: Multiple Choice (graphing calculator required)

    Questions
    13
    Time
    40 min
    Weight
    18.8%

    Calculator allowed

    Format details

    Four answer choices (A-D). A graphing calculator is required for some of these questions (solving equations, intersections, regressions, evaluating models), not all. Bluebook provides a built-in Desmos graphing calculator; approved handhelds are also allowed. Radian mode. Numeric choices are usually given to three decimal places. New for May 2027: 13 questions (was 12).

  • Section II, Part A: Free Response (graphing calculator required)

    Questions
    2
    Time
    35 min
    Weight
    18.8%

    Calculator allowed

    Format details

    Task types: Function Concepts, Modeling a Non-Periodic Context

    Question 1 (Function Concepts) and Question 2 (Modeling a Non-Periodic Context), 6 points each. Viewed in Bluebook, answered by hand in a paper booklet. Decimal answers must be accurate to three places after the decimal point. New for May 2027: 35 minutes (was 30), and Question 2 has a new structure.

  • Section II, Part B: Free Response (no calculator)

    Questions
    2
    Time
    35 min
    Weight
    18.8%

    No calculator

    Format details

    Task types: Modeling a Periodic Context, Symbolic Manipulations

    Question 3 (Modeling a Periodic Context) and Question 4 (Symbolic Manipulations), 6 points each. Exact values are required wherever they can be found without a calculator. New for May 2027: 35 minutes (was 30).

How the 1 to 5 score is set

Section I: one point per correct multiple-choice answer, no penalty for wrong answers; the 29 no-calculator questions are 43.75% of the composite and the 13 calculator questions are 18.75%. Section II: four 6-point questions scored analytically, one point per scoring row (A1, A2, B1, B2, C1, C2), each question 9.375% of the composite (37.5% in all). Some parts offer partial credit (one of the two points) when neither point is fully earned. In calculator questions, the first decimal presentation error (fewer than three decimal places, or an exact value where a decimal was asked) costs that point; later ones in the same question do not. The weighted composite is converted to the 1-5 AP score with cut points set each year.

What you bring and get

No formula sheet or reference sheet is provided: every identity, formula, and unit-circle value must be known. A graphing calculator is required for Section I Part B and Section II Part A; Bluebook includes a built-in Desmos graphing calculator, and approved handheld graphing calculators may be used. Calculators must be in radian mode. Multiple-choice questions are answered on screen in Bluebook; free-response questions are read in Bluebook and answered by hand in a paper booklet.

Skills the exam scores

  • 1.ASolve equations and inequalities

    Practice 1, Procedural and Symbolic Fluency (35-50% of the exam). Solve equations and inequalities given analytically, with and without technology: polynomial and rational inequalities, exponential and logarithmic equations, trigonometric equations on an interval or in general.

  • 1.BRewrite in equivalent forms

    Practice 1, Procedural and Symbolic Fluency. Express functions, equations, or expressions in analytically equivalent forms that are useful for a purpose: factored form to show zeros, a single logarithm, a single power of one base, an identity-simplified trig expression.

  • 1.CConstruct new functions

    Practice 1, Procedural and Symbolic Fluency. Build new functions with transformations, compositions, inverses, or regressions to fit a context, stated criteria, or data, with and without technology, including solving for model parameters.

  • 2.AIdentify information from representations

    Practice 2, Multiple Representations (20-30% of the exam). Read the information needed to answer a question or build a model from a graph, a table, a formula, or a verbal description, with and without technology.

  • 2.BConstruct equivalent representations

    Practice 2, Multiple Representations. Produce an equivalent graph, table, formula, or verbal description of a function that is useful in a given setting, such as sketching a sinusoid from a description or labeling key points on a graph.

  • 3.ADescribe function characteristics

    Practice 3, Communication and Reasoning (30-40% of the exam). Describe increasing and decreasing behavior, concavity, extrema, zeros, asymptotes, and end behavior with the precision the representation allows, including correct limit notation.

  • 3.BApply numerical results in context

    Practice 3, Communication and Reasoning. Use and interpret a numerical result in its context, with correct units: what an average rate of change or a solved input value means for the situation.

  • 3.CSupport conclusions with reasoning

    Practice 3, Communication and Reasoning. Justify a choice or conclusion with a logical rationale or appropriate data: why a function type fits a table, why a function is invertible, why a model must be restricted to a domain.