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Algebra and Function Foundations

A lean diagnostic on-ramp covering the algebraic grammar precalculus assumes: real-number properties and notation, exponent and radical fluency, rewriting polynomial and rational expressions, and solving equations and inequalities. Topics: Real Numbers and Algebraic Properties, Exponents and Radicals, Polynomial and Rational Expressions, Solving Equations and Inequalities.

Functions and Their Graphs

The central object of the course: the coordinate plane and graphs of equations, lines and their slopes and forms, the definition and notation of functions, reading graph features, the library of parent functions, transformations, combinations and composition, and inverse functions. Topics: Coordinate Plane and Graphs, Linear Equations in Two Variables, Functions and Function Notation, Analyzing Graphs of Functions, Library of Parent Functions, Transformations of Functions, Combinations and Composition of Functions, Inverse Functions.

Polynomial and Rational Functions

Functions defined by polynomials and ratios of polynomials: quadratics and their graphs, the end behavior and zeros of higher-degree polynomials, polynomial division and the zero theorems, complex numbers, and the graphs, asymptotes, and inequalities of rational functions. Topics: Quadratic Functions, Polynomial Functions of Higher Degree, Polynomial and Synthetic Division, Complex Numbers, Zeros of Polynomial Functions, Rational Functions, Nonlinear Inequalities.

Exponential and Logarithmic Functions

The inverse pair of exponential and logarithmic functions: their graphs and key features, the definition of the logarithm, the properties used to rewrite logarithmic expressions, solving exponential and logarithmic equations, and growth, decay, and other models. Topics: Exponential Functions and Graphs, Logarithmic Functions and Graphs, Properties of Logarithms, Exponential and Logarithmic Equations, Exponential and Logarithmic Models.

Trigonometry

Trigonometry from angles to applications: angle measure in degrees and radians, the unit-circle and right-triangle definitions of the six trigonometric functions, trig functions of any angle, the graphs of all six functions and their inverses, and right-triangle and periodic applications. Topics: Angles and Their Measure, Trigonometric Functions: The Unit Circle, Right Triangle Trigonometry, Trigonometric Functions of Any Angle, Graphs of Sine and Cosine, Graphs of Other Trigonometric Functions, Inverse Trigonometric Functions, Applications and Models.

Analytic Trigonometry

Working symbolically with trigonometric expressions and equations: simplifying with the fundamental identities, verifying identities, solving trigonometric equations, and applying the sum and difference, double-angle, half-angle, and product-to-sum formulas. Topics: Using Fundamental Identities, Verifying Trigonometric Identities, Solving Trigonometric Equations, Sum and Difference Formulas, Multiple-Angle and Product-to-Sum Formulas.

Applications of Trigonometry

Trigonometry beyond the right triangle: solving oblique triangles with the Laws of Sines and Cosines, vector operations and the dot product in the plane, and the trigonometric (polar) form of complex numbers with DeMoivre's Theorem. Topics: Law of Sines, Law of Cosines, Vectors in the Plane, Vectors and Dot Products, Trigonometric Form of Complex Numbers.

Systems of Equations and Inequalities

Solving systems of equations and inequalities: linear and nonlinear systems by substitution, elimination, and Gaussian methods, decomposing rational expressions into partial fractions, and graphing systems of inequalities and solving linear programming problems. Topics: Two-Variable Systems, Multivariable Linear Systems, Partial Fractions, Systems of Inequalities and Linear Programming.

Matrices and Determinants

Matrix methods for linear systems: representing and row-reducing augmented matrices, matrix algebra, inverses for solving systems, determinants, and Cramer's Rule. Topics: Matrices and Systems of Equations, Operations with Matrices, Inverse of a Square Matrix, Determinants and Cramer's Rule.

Sequences, Series, and Probability

Discrete patterns and counting: sequence and series notation, arithmetic and geometric sequences and their sums, mathematical induction, the Binomial Theorem, counting principles, and basic probability. Topics: Sequences and Series, Arithmetic Sequences and Series, Geometric Sequences and Series, Mathematical Induction, The Binomial Theorem, Counting and Probability.

Analytic Geometry: Conics, Parametric, and Polar

Curves in the plane beyond function graphs: the conic sections and their equations, rotation of axes, parametric representations of curves, and polar coordinates with the graphs and conic equations they enable. Topics: Parabolas, Ellipses, Hyperbolas, Rotation of Conics, Parametric Equations, Polar Coordinates and Graphs.

Limits and the On-Ramp to Calculus

An introduction to the limit concept that motivates calculus: estimating limits graphically and numerically, evaluating them algebraically, and using limits to find instantaneous rate of change as the slope of a tangent line and to approximate the area under a curve. Topics: Introduction to Limits, Limits and Rates of Change.

Polynomial and Rational Functions

AP Exam weighting: 30-40%. Topics: 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 of Polynomial and Rational Expressions, Transformations of Functions, Function Model Selection and Assumption Articulation, Function Model Construction and Application.

Exponential and Logarithmic Functions

AP Exam weighting: 27-40%. Topics: 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.

Trigonometric and Polar Functions

AP Exam weighting: 30-35%. Topics: 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, Trigonometry and Polar Coordinates, Polar Function Graphs, Rates of Change in Polar Functions.

Functions Involving Parameters, Vectors, and Matrices

AP Exam weighting: not assessed on the AP Exam. Topics: 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.