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AP Precalculus Study Guide

The whole course, unit by unit

Lessons, worked examples, diagrams, and practice PDFs for all four units. The AP exam assesses Units 1–3; Unit 4 is additional course content.

Four units

Polynomial and Rational Functions; Exponential and Logarithmic Functions; Trigonometric and Polar Functions; then Functions Involving Parameters, Vectors, and Matrices.

Three are assessed

The AP exam covers Units 1 through 3 only. Unit 4 is part of the course but never appears on the exam.

Grade levels

Usually taken in grades 10 through 12, after Algebra 2, as preparation for AP Calculus or AP Statistics.

Graphing calculator

Required. Part of the exam assumes one, and several question types are written to be impractical without it.

Units

The four units

1

Unit 1 · Largest assessed unit

Polynomial and Rational Functions

The unit that establishes the language for the rest of the course. It introduces rate of change as the thing that distinguishes function families, then works through polynomial behavior and the more delicate business of rational functions, where the graph depends on structure that is invisible until you factor.

  1. 1.1 Average Rate of Change
  2. 1.2 Successive Differences and Degree
  3. 1.3 Zeros and Multiplicity
  4. 1.4 End Behavior
  5. 1.5 Vertical Asymptotes and Holes
  6. 1.6 End Behavior of Rational Functions
  7. 1.7 Transformations of Functions
Average rate of changeSuccessive differencesMultiplicityEnd behaviorVertical asymptoteHole
Read Unit 1Unit MCQ Practice
2

Unit 2 · Moderate

Exponential and Logarithmic Functions

A short unit with a single organising idea: exponential functions change by equal ratios over equal input intervals, and logarithms are the inverse that recovers the exponent. Most of the unit is consequences of that sentence, including modeling, equation solving, and the semi-log plot.

  1. 2.1 Constant Ratios and Model Choice
  2. 2.2 Growth and Decay Factors
  3. 2.3 Rescaling the Exponent
  4. 2.4 Logarithms as Inverses
  5. 2.5 Logarithm Properties
  6. 2.6 Solving Exponential and Logarithmic Equations
  7. 2.7 Semi-Log Plots
Constant ratioGrowth factorDecay factorLogarithmExtraneous solutionSemi-log plot
Read Unit 2Unit MCQ Practice
3

Unit 3 · Large assessed unit

Trigonometric and Polar Functions

The unit that introduces periodicity. It builds the unit circle, turns it into sinusoidal functions with four adjustable parameters, uses those to model repeating phenomena, and then changes coordinate systems entirely to look at polar functions.

  1. 3.1 Periodic Phenomena
  2. 3.2 The Unit Circle
  3. 3.3 Sinusoidal Parameters
  4. 3.4 Phase Shift
  5. 3.5 Modeling with Sinusoids
  6. 3.6 Trigonometric Equations and Identities
  7. 3.7 Polar Functions
PeriodMidlineAmplitudePhase shiftReference angleUnit circle
Read Unit 3Unit MCQ Practice
4

Unit 4 · Not assessed on the exam

Functions Involving Parameters, Vectors, and Matrices

Taught as part of the course but never tested. It covers parametric and implicitly defined curves, vectors, and matrices as transformations. Worth learning for later courses in calculus and linear algebra, but it should not consume revision time in the weeks before the exam.

  1. 4.1 Parametric Equations
  2. 4.2 Implicitly Defined Curves
  3. 4.3 Vectors
  4. 4.4 Matrix Operations
  5. 4.5 Determinants and Invertibility
  6. 4.6 Linear Transformations
ParameterImplicit equationVector magnitudeComponentDeterminantInvertible
Read Unit 4Unit MCQ Practice

Curriculum scope

What this sequence includes

This is the College Board AP Precalculus framework, not a generic precalculus syllabus. The organising idea across all four units is that a function family is identified by how its output changes, and that a claim about a function must be justified from an equation, a graph, a table, or a context rather than asserted.

Algebra and trigonometry core

Essential foundations

  • • Rates of change, including average rate of change over an interval and the behavior of successive differences
  • • Polynomial functions: zeros, multiplicity, end behavior, and the effect of degree
  • • Rational functions: vertical asymptotes, holes, and end-behavior comparison of degrees
  • • Exponential functions as constant multiplicative change, and logarithms as their inverse
  • • Trigonometric functions built from the unit circle, and sinusoidal modeling of periodic phenomena
  • • Polar functions and how a changing radius moves a curve toward or away from the pole

Breadth beyond the assessed core

Included in this sequence

  • • Function transformations, composition, inverses, domain, and range across every family
  • • Model selection from data, including the use of semi-log plots to identify exponential behavior
  • • Parametric equations and implicitly defined curves
  • • Vectors, matrix operations, determinants, and linear transformations

Varies by course

Compare with the school syllabus

  • Unit 4 depth. Because it is not assessed, schools treat parameters, vectors, and matrices very differently: some teach it fully, some compress it, and some move it after the exam.
  • Calculator model. Course expectations differ by school, but the exam requires a graphing calculator on the sections that permit one.
  • Prior trigonometry. Students arriving from an Algebra 2 course that already covered the unit circle will find Unit 3 lighter than those meeting it for the first time.
  • Limits. AP Precalculus stops short of limits. A school course that adds them is going beyond the framework, and they will not be assessed.

Glossary

Terms worth being precise about

Average rate of change

The change in output divided by the change in input across an interval, equal to the slope of the line joining the endpoints.

Multiplicity

The exponent on a factor at a zero, which decides whether the graph crosses the axis or touches and turns.

End behavior

What a function does as the input grows without bound in either direction, determined by the leading term alone.

Hole

A single missing point on a rational graph, produced by a factor that cancels from both numerator and denominator.

Vertical asymptote

A line the graph approaches without meeting, produced by a denominator factor that does not cancel.

Pre-image

A point on the original graph of y = f(x), before a transformation is applied.

Image

The point a pre-image is mapped to once a transformation is applied, so the transformed graph is the set of images of the original.

Growth factor

The constant an exponential model multiplies by over one period, equal to one plus the growth rate.

Semi-log plot

A plot with a logarithmic vertical axis, on which exponential data appears as a straight line.

Midline

The horizontal line halfway between the maximum and minimum of a sinusoidal function.

Amplitude

Half the distance between the maximum and the minimum of a sinusoidal function.

Phase shift

The horizontal translation of a sinusoid, read only after the coefficient of the variable has been factored out.

Pole

The origin of a polar coordinate system, from which the radius is measured.

Parameter

A third variable that generates the coordinates of a curve and records how the curve is traversed.

Determinant

A number computed from a square matrix that is zero exactly when the matrix is not invertible.

FRQ practice & solutions

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