AP Precalculus (College Board): complete guide to the units, the mathematical practices and the exam

College Board AP Precalculus is designed to prepare students for college-level calculus and other courses that rely on the analysis of functions. The course is built on the big idea that functions model the world, together with a set of mathematical practices, and the content is organized into units around families of functions. There is no coursework; computation, reasoning and justification are examined directly in both sections of the exam. This page is the index: below is a map of the units, the exam structure, and how to study each one. This library now covers the full course, Units 1 to 4.

The AP Precalculus units

The College Board organizes the content into four units. Units 1 to 3 are assessed on the exam; Unit 4 is required content that prepares students for later study.

Unit 1 Polynomial and Rational Functions (about 30 to 40%)

Change in tandem and rates of change, linear and quadratic rate behavior, polynomial functions with their complex zeros and end behavior, rational functions with their zeros, vertical asymptotes and holes, equivalent representations, transformations, and function model selection and construction.

Unit 2 Exponential and Logarithmic Functions (about 27 to 40%)

Arithmetic and geometric sequences, the contrast between linear and exponential change, exponential functions and their manipulation, exponential modelling and competing-model validation, composition and inverses, logarithmic expressions and functions, logarithmic manipulation, exponential and logarithmic equations and inequalities, logarithmic modelling, and semi-log plots.

Unit 3 Trigonometric and Polar Functions (about 30 to 35%)

The unit circle, sine, cosine and tangent functions and their graphs, transformations and inverses of trigonometric functions, trigonometric equations and identities, and polar functions.

Unit 4 Functions Involving Parameters, Vectors, and Matrices

Parametric functions, vectors, matrices and their transformations. This unit is required content but is not assessed on the AP Exam.

Exam structure

The AP Precalculus exam is 3 hours and has two equally weighted sections. A graphing calculator is required on the designated parts and forbidden on the others. The exam assesses Units 1 to 3.

• Section I, multiple choice - 50%. A no-calculator part and a calculator part.
• Section II, free response - 50%. A calculator part and a no-calculator part.

The free-response questions are written from the mathematical practices, so they ask you to compute, connect graphical, numerical and analytical representations, justify conclusions with definitions and properties, and use correct notation throughout.

How to study AP Precalculus

AP Precalculus rewards fluent algebra, conceptual understanding of function behavior, and clear justification.

1. Work from the Course and Exam Description. Each topic (for example 2.8 Inverse Functions) maps to specific learning objectives and essential knowledge statements that exam questions are written from.
2. Learn the practices, not just the answers. Practice justifying with definitions and properties, translating among representations, and writing correct notation, because the free-response questions are scored on these skills.
3. Master the no-calculator algebra. Factoring, evaluating logarithms, finding inverses, and solving exponential and logarithmic equations all recur on the no-calculator parts.
4. Connect the function families. Geometric sequences become exponential functions; exponentials and logarithms are inverses; rates of change foreshadow calculus. Seeing the connections makes the rules feel inevitable.
5. Rehearse both calculator and no-calculator timing. Know which tools are allowed where, and practice presenting setups and justifications even when the calculator does the arithmetic or the regression.

The units, topic by topic

Each topic has a Course-and-Exam-Description-level answer page with worked exam questions and cross-links, plus an overview guide and quiz. Browse the set at /ap/precalculus/syllabus. The library now covers all four units:

• Unit 1: change in tandem, rates of change, rates of change in linear and quadratic functions, polynomial functions and rates of change, polynomial functions and complex zeros, polynomial functions and end behavior, rational functions and end behavior, rational functions and zeros, rational functions and vertical asymptotes, rational functions and holes, equivalent representations, transformations of functions, function model selection, function model construction.
• Unit 2: change in arithmetic and geometric sequences, change in linear and exponential functions, exponential functions, exponential function manipulation, exponential function context and data modeling, competing function model validation, composition of functions, inverse functions, logarithmic expressions, inverses of exponential functions, logarithmic functions, logarithmic function manipulation, exponential and logarithmic equations and inequalities, logarithmic function context and data modeling, semi-log plots.
• Unit 3: periodic phenomena, sine, cosine and tangent, sine and cosine function values, sine and cosine function graphs, sinusoidal functions, sinusoidal function transformations, sinusoidal function context and data modeling, the tangent function, inverse trigonometric functions, trigonometric equations and inequalities, the secant, cosecant and cotangent functions, equivalent representations of trigonometric functions, polar coordinates, polar function graphs, rates of change in polar functions.
• Unit 4: parametric functions, parametric functions modeling planar motion, parametric functions and rates of change, parametrically defined circles and lines, implicitly defined functions, conic sections, parametrization of implicitly defined functions, vectors, vector-valued functions, matrices, the inverse and determinant of a matrix, linear transformations and matrices, matrices as functions, matrices modeling contexts. Unit 4 is required content that is not assessed on the AP Exam.

For the official Course and Exam Description

The College Board publishes the full Course and Exam Description, released free-response questions, scoring guidelines and sample questions at apcentral.collegeboard.org. Always study from the current Course and Exam Description and the College Board's own released exams, because question style and the mathematical practices are board-specific.