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Chapters

This textbook is organized into 23 chapters covering 307 concepts across 4 units, following the College Board AP Pre-Calculus curriculum framework.

Unit 1: Polynomial and Rational Functions

  1. Mathematical Foundations - Core building blocks including the real number system, coordinate plane, variables, expressions, and notation systems.
  2. Functions and Their Properties - Function definition, notation, evaluation, domain, range, graphing, intercepts, behavior, and symmetry.
  3. Rates of Change - How quantities change together, average rate of change, slope, secant lines, concavity, and increasing/decreasing rates.
  4. Linear Functions - Linear function forms (slope-intercept, point-slope, standard), parallel and perpendicular lines, and linear regression.
  5. Quadratic Functions and Complex Numbers - Quadratic forms, vertex, axis of symmetry, completing the square, quadratic formula, discriminant, and complex numbers.
  6. Polynomial Fundamentals - Polynomial definitions, degree, leading coefficient, operations, factoring, division, and key theorems.
  7. Polynomial Zeros and Graphs - Zeros, multiplicity, end behavior, turning points, extrema, graphing, and the Fundamental Theorem of Algebra.
  8. Rational Functions - Rational expressions, asymptotes, holes, zeros, domain, end behavior, graphing, and inequalities.

Transition: Transformations and Modeling

  1. Function Transformations - Parent functions, translations, stretches, compressions, reflections, combined transformations, and writing transformed equations.
  2. Data Modeling and Regression - Model selection, data analysis, regression, validation, residuals, goodness of fit, and competing models.

Unit 2: Exponential and Logarithmic Functions

  1. Composition and Inverse Functions - Function composition, domain of compositions, decomposition, inverse functions, one-to-one functions, and horizontal line test.
  2. Sequences - Arithmetic and geometric sequences, common difference/ratio, recursive and explicit formulas, convergence and divergence.
  3. Exponential Functions - Exponential growth/decay, natural base e, compound interest, continuous growth, equations, graphs, transformations, and modeling.
  4. Logarithmic Functions - Logarithm definition, properties, equations, graphs, transformations, modeling, semi-log plots, and data linearization.

Unit 3: Trigonometric and Polar Functions

  1. Angles and the Unit Circle - Angle measures, radian/degree conversion, arc length, sector area, standard position, reference angles, unit circle, and special angles.
  2. Trigonometric Functions - Sine, cosine, tangent, reciprocal functions, evaluation, exact values, right triangle trigonometry, and SOH-CAH-TOA.
  3. Trigonometric Graphs - Sine/cosine/tangent graphs, amplitude, period, midline, frequency, phase shift, sinusoidal functions, and reciprocal function graphs.
  4. Sinusoidal Modeling - Graphing trig transformations, sinusoidal modeling, periodic data analysis, real-world models, and sinusoidal regression.
  5. Trigonometric Identities - Pythagorean, reciprocal, quotient, cofunction, even-odd identities, sum/difference and double angle formulas, verifying and simplifying.
  6. Inverse Trigonometric Functions and Equations - Arcsine, arccosine, arctangent, inverse trig domains/ranges/graphs, solving trig equations, general solutions, and inequalities.

Unit 4: Functions Involving Parameters, Vectors, and Matrices

  1. Polar and Parametric Functions - Polar coordinates, conversions, polar graphs, parametric equations, motion, and projectile models.
  2. Vectors - Vector definition, notation, magnitude, direction, component form, unit vectors, operations, dot product, and applications.
  3. Matrices and Exam Preparation - Matrix operations, determinant, inverse, linear transformations, applications, and AP exam strategies.

How to Use This Textbook

Chapters are sequenced to respect concept dependencies — each chapter builds on concepts from earlier chapters. Students should progress through the chapters in order, though Unit 4 content is not tested on the AP exam and may be studied independently. Each chapter includes a list of concepts covered and links to prerequisite chapters.

Note: Each chapter includes a list of concepts covered. Make sure to complete prerequisites before moving to advanced chapters.