Skip to content

AP Pre-Calculus Course Description

Overview

An interactive intelligent textbook for AP Pre-Calculus designed to prepare students for AP Pre-Calculus tests. This course follows the College Board AP Pre-Calculus curriculum framework, which builds a deep understanding of functions and their properties as essential preparation for calculus and other college-level mathematics courses.

AP Pre-Calculus was introduced by the College Board in the 2023–2024 school year to create a nationally standardized pre-calculus course that emphasizes function modeling, reasoning with functions, and making connections across multiple representations.

This course features a large collection of Micro-Simulations (MicroSims) that are used by the students to visualize complex topics.

Target Audience

  • Primary Audience: High school students (typically grades 10–12) preparing for the AP Pre-Calculus exam and future AP Calculus coursework
  • Prerequisites: Successful completion of Algebra 2 or equivalent, including familiarity with linear and quadratic functions, basic polynomial operations, systems of equations, and foundational coordinate geometry
  • Reading Level: Senior High School (grades 10–12)

Course Structure

The course is organized into four units following the College Board AP Pre-Calculus framework. Units 1–3 are tested on the AP exam. Unit 4 is enrichment content not included on the AP exam but valuable for calculus readiness.

Unit Topic Key Concepts
1 Polynomial and Rational Functions Rates of change, polynomial behavior, rational functions, end behavior, zeros, asymptotes, transformations
2 Exponential and Logarithmic Functions Exponential growth/decay, logarithmic functions, compositions, inverses, data modeling, semi-log plots
3 Trigonometric and Polar Functions Periodic phenomena, unit circle, sine/cosine/tangent, inverse trig, polar coordinates, polar function graphs
4 Functions Involving Parameters, Vectors, and Matrices (Not on AP Exam) Parametric functions, vectors, matrices, linear transformations

Topics Covered

Unit 1: Polynomial and Rational Functions

  1. Change in Tandem — how two quantities change together
  2. Rates of Change — average rates of change and their meaning
  3. Rates of Change in Linear and Quadratic Functions — constant vs. changing rates
  4. Polynomial Functions and Rates of Change — higher-degree polynomial behavior
  5. Polynomial Functions and Complex Zeros — zeros over real and complex numbers
  6. Polynomial Functions and End Behavior — leading term dominance
  7. Rational Functions and End Behavior — horizontal and slant asymptotes
  8. Rational Functions and Zeros — numerator zeros, holes, and vertical asymptotes
  9. Rational Functions and Vertical Asymptotes — unbounded behavior near excluded values
  10. Transformations of Functions — translations, reflections, dilations
  11. Function Model Selection and Assumption Articulation — choosing appropriate models
  12. Function Model Construction and Application — building models from data

Unit 2: Exponential and Logarithmic Functions

  1. Change in Arithmetic and Geometric Sequences — additive vs. multiplicative patterns
  2. Change in Linear and Exponential Functions — constant rate vs. proportional rate
  3. Exponential Functions — base, growth factor, and decay factor
  4. Exponential Function Manipulation — equivalent forms and properties
  5. Exponential Function Context and Data Modeling — real-world applications
  6. Competing Function Model Validation — comparing model fits
  7. Composition of Functions — combining functions
  8. Inverse Functions — reversing input-output relationships
  9. Logarithmic Expressions — definition and properties of logarithms
  10. Logarithmic Functions — graphs, domains, ranges
  11. Logarithmic Function Context and Data Modeling — linearizing exponential data
  12. Semi-Log Plots — interpreting data on logarithmic scales

Unit 3: Trigonometric and Polar Functions

  1. Periodic Phenomena — recognizing repeating patterns
  2. Sine, Cosine, and Tangent — definitions from the unit circle
  3. Sine and Cosine Function Values — special angles and reference angles
  4. Sine and Cosine Function Graphs — amplitude, period, midline
  5. Sinusoidal Functions — modeling with sine and cosine
  6. Sinusoidal Function Context and Data Modeling — real-world periodic data
  7. Sinusoidal Function Transformations — phase shift, vertical shift, amplitude, period
  8. The Tangent Function — definition, graph, and properties
  9. Inverse Trigonometric Functions — arcsine, arccosine, arctangent
  10. Trigonometric Equations and Inequalities — solving for angles
  11. The Secant, Cosecant, and Cotangent Functions — reciprocal trig functions
  12. Equivalent Representations of Trigonometric Functions — identities and co-functions
  13. Trigonometry and Polar Coordinates — converting between coordinate systems
  14. Polar Function Graphs — roses, limaçons, circles, cardioids
  15. Rates of Change in Polar Functions — how polar curves change

Unit 4: Functions Involving Parameters, Vectors, and Matrices (Not on AP Exam)

  1. Parametric Functions — expressing x and y as functions of a parameter
  2. Parametric Functions Modeling Planar Motion — position, velocity, direction
  3. Parametric Functions and Rates of Change — derivatives of parametric curves
  4. Vectors — magnitude, direction, component form
  5. Vector-Valued Functions — functions that return vectors
  6. Matrices — operations, dimensions, properties
  7. Matrices as Functions and Linear Transformations — geometric transformations using matrices
  8. Matrices Modeling Contexts — applications of matrix operations

Topics NOT Covered

To maintain focus on pre-calculus foundations, this course does not cover:

  • Limits — formal definition and evaluation of limits (covered in AP Calculus)
  • Derivatives — instantaneous rates of change and differentiation rules (covered in AP Calculus)
  • Integrals — antiderivatives and area under curves (covered in AP Calculus)
  • Formal proofs — rigorous mathematical proofs (covered in discrete mathematics)
  • Statistics and probability — data analysis and inference (covered in AP Statistics)
  • Conic sections in depth — parabolas, ellipses, hyperbolas beyond what appears in function contexts
  • Series and sequences beyond arithmetic/geometric — convergence, Taylor series (covered in AP Calculus BC)
  • Linear algebra beyond Unit 4 basics — eigenvalues, vector spaces (covered in college linear algebra)
  • Differential equations — solving equations involving derivatives (covered in post-calculus courses)

Learning Objectives (2001 Bloom's Taxonomy)

Level 1: Remember

  1. Recall the definitions of polynomial, rational, exponential, logarithmic, and trigonometric functions
  2. State the properties of exponents and logarithms
  3. List the special angle values for sine, cosine, and tangent on the unit circle
  4. Identify the standard forms of polynomial, exponential, and sinusoidal functions
  5. Define key vocabulary: asymptote, zero, end behavior, period, amplitude, midline, domain, range

Level 2: Understand

  1. Explain the relationship between average rate of change and the behavior of a function over an interval
  2. Describe how the degree and leading coefficient of a polynomial determine its end behavior
  3. Interpret the meaning of zeros, vertical asymptotes, and holes in rational functions
  4. Explain the relationship between exponential and logarithmic functions as inverses
  5. Describe how the parameters of a sinusoidal function (amplitude, period, phase shift, midline) affect its graph
  6. Explain the connection between the unit circle and the graphs of trigonometric functions
  7. Interpret polar coordinates and describe how polar graphs differ from Cartesian graphs
  8. Explain the distinction between arithmetic and geometric sequences in terms of additive vs. multiplicative change

Level 3: Apply

  1. Calculate average rates of change for polynomial, rational, exponential, and trigonometric functions over specified intervals
  2. Graph polynomial functions using zeros, multiplicity, and end behavior
  3. Graph rational functions using zeros, vertical asymptotes, holes, and horizontal/slant asymptotes
  4. Apply transformation rules (translations, reflections, dilations) to function graphs
  5. Convert between exponential and logarithmic forms of equations
  6. Use composition of functions to build new functions from existing ones
  7. Find inverse functions algebraically and verify the inverse relationship
  8. Evaluate trigonometric functions using the unit circle and reference angles
  9. Solve trigonometric equations for specific angle values within given domains
  10. Convert between rectangular and polar coordinates
  11. Graph polar functions including roses, limaçons, and cardioids
  12. Use parametric equations to model planar motion
  13. Perform matrix operations including addition, multiplication, and scalar multiplication

Level 4: Analyze

  1. Analyze the rate of change of a function to determine whether it is increasing, decreasing, concave up, or concave down over an interval
  2. Compare and contrast polynomial, rational, exponential, and trigonometric function models for a given data set
  3. Determine which function type best models a given real-world scenario based on the pattern of change
  4. Analyze the relationship between a function and its inverse, including domain and range restrictions
  5. Distinguish between situations modeled by linear vs. exponential growth and justify the choice
  6. Analyze how changes in parameters of a sinusoidal function correspond to changes in the real-world phenomenon being modeled
  7. Examine semi-log plots to determine whether data exhibits exponential behavior
  8. Analyze the behavior of polar functions by examining how the radius changes as the angle varies

Level 5: Evaluate

  1. Assess the validity of a proposed function model by comparing predicted vs. actual values
  2. Evaluate competing function models (polynomial, exponential, logarithmic, sinusoidal) and justify which provides the best fit for a data set
  3. Judge whether a function model remains appropriate outside its original domain of data collection
  4. Critique claims about function behavior based on evidence from graphs, tables, and algebraic representations
  5. Evaluate the reasonableness of solutions to trigonometric equations within real-world contexts
  6. Assess the strengths and limitations of different coordinate systems (rectangular vs. polar) for representing specific curves

Level 6: Create

  1. Construct a piecewise-defined function to model a real-world situation with different behaviors over different intervals
  2. Build a sinusoidal model from real-world periodic data (e.g., tides, temperatures, daylight hours)
  3. Develop an exponential or logarithmic model from raw data, including determining appropriate parameters
  4. Create a rational function model with specified zeros, asymptotes, and intercepts to match given constraints
  5. Design a polar equation to produce a curve with specified symmetry and shape properties
  6. Formulate a parametric model to represent the motion of an object in the plane given initial conditions

AP Exam Format

The AP Pre-Calculus exam consists of:

  • Section I: Multiple Choice (62.5% of score)
    • Part A: 28 questions, 80 minutes (no calculator)
    • Part B: 12 questions, 40 minutes (graphing calculator required)
  • Section II: Free Response (37.5% of score)
    • Part A: 2 questions, 30 minutes (graphing calculator required)
    • Part B: 2 questions, 30 minutes (no calculator)

Only content from Units 1–3 appears on the AP exam. Unit 4 is designated as non-AP exam content.

Assessment Within This Textbook

Learning is assessed through:

  • Chapter Quizzes — Multiple-choice questions aligned to specific learning objectives at the end of each chapter
  • Interactive MicroSims — Hands-on explorations that build conceptual understanding through dynamic visualizations
  • Practice Problems — Exercises progressing from basic skills to AP-level application and analysis

Time Commitment

  • Estimated Duration: Full academic year (approximately 140–160 instructional hours)
  • Chapters: 23 chapters across 4 units
  • Pace: Self-paced when used as supplementary material; follows the school calendar when used as a primary text

Resources

  • Primary Text: This interactive intelligent textbook
  • Supplementary: College Board AP Pre-Calculus Course and Exam Description (CED), Khan Academy, Desmos graphing calculator
  • Community: GitHub Issues for questions and feedback

References

College Board AP Precalculus Course and Exam Description