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AP Precalculus

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AP Precalculus covers polynomial and rational functions, exponential and logarithmic functions, trigonometric and polar functions, and an introduction to vectors, matrices and parametric functions. It is the bridge into calculus, and the exam is unusually strict about how you describe a function.

Describing change is the whole course. Expect to say where a function increases or decreases, how fast, and whether that rate is itself increasing — in a full sentence, with the correct vocabulary.

Learn the parent functions and their transformations cold. Everything else in the course is built from them, and recognising a shifted or stretched parent instantly is what makes the harder questions short.

Notes follow the four CED units — polynomial and rational, exponential and logarithmic, trigonometric and polar, and functions involving parameters — each with worked examples. Sample exam questions are in the library. Precalculus asks you to describe how a function behaves in sentences, which the examples model.

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Past Papers
  • 1

    Polynomial and Rational Functions

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    • 1.1 Change in Tandem
    • 1.2 Rates of Change
    • 1.3 Rates of Change in Linear and Quadratic Functions
    • 1.4 Polynomial Functions and Rates of Change
    • 1.5 Polynomial Functions and Complex Zeros
    • 1.6 Polynomial Functions and End Behavior
    • 1.7 Rational Functions and End Behavior
    • 1.8 Rational Functions and Zeros
    • 1.9 Rational Functions and Vertical Asymptotes
    • 1.10 Rational Functions and Holes
    • 1.11 Equivalent Representations of Polynomial and Rational Expressions
    • 1.12 Transformations of Functions
    • 1.13 Function Model Selection and Assumption Articulation
    • 1.14 Function Model Construction and Application
    • 1.14 Exam tips
  • 2

    Exponential and Logarithmic Functions

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    • 2.1 Change in Arithmetic and Geometric Sequences
    • 2.2 Change in Linear and Exponential Functions
    • 2.3 Exponential Functions
    • 2.4 Exponential Function Manipulation
    • 2.5 Exponential Function Context and Data Modeling
    • 2.6 Competing Function Model Validation
    • 2.7 Composition of Functions
    • 2.8 Inverse Functions
    • 2.9 Logarithmic Expressions
    • 2.10 Inverses of Exponential Functions
    • 2.11 Logarithmic Functions
    • 2.12 Logarithmic Function Manipulation
    • 2.13 Exponential and Logarithmic Equations and Inequalities
    • 2.14 Logarithmic Function Context and Data Modeling
    • 2.15 Semi-log Plots
    • 2.15 Exam tips
  • 3

    Trigonometric and Polar Functions

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    • 3.1 Periodic Phenomena
    • 3.2 Sine, Cosine, and Tangent
    • 3.3 Sine and Cosine Function Values
    • 3.4 Sine and Cosine Function Graphs
    • 3.5 Sinusoidal Functions
    • 3.6 Sinusoidal Function Transformations
    • 3.7 Sinusoidal Function Context and Data Modeling
    • 3.8 The Tangent Function
    • 3.9 Inverse Trigonometric Functions
    • 3.10 Trigonometric Equations and Inequalities
    • 3.11 The Secant, Cosecant, and Cotangent Functions
    • 3.12 Equivalent Representations of Trigonometric Functions
    • 3.13 Trigonometry and Polar Coordinates
    • 3.14 Polar Function Graphs
    • 3.15 Rates of Change in Polar Functions
    • 3.15 Exam tips
  • 4

    Functions Involving Parameters, Vectors, and Matrices

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    • 4.1 Parametric Functions
    • 4.2 Parametric Functions and Planar Motion
    • 4.3 Parametric Functions and Rates of Change
    • 4.4 Parametrically Defined Circles and Lines
    • 4.5 Implicitly Defined Functions
    • 4.6 Conic Sections
    • 4.7 Parametrizing Implicit Curves
    • 4.8 Vectors
    • 4.9 Vector-Valued Functions
    • 4.10 Matrices
    • 4.11 The Inverse and Determinant of a Matrix
    • 4.12 Linear Transformations and Matrices
    • 4.13 Matrices as Functions
    • 4.14 Matrices Modeling Contexts
    • 4.14 Exam tips

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IGCSE, A-Level & AP