AP Precalculus Course Overview

This guide summarizes the structure, major units, and essential topics of the AP Precalculus curriculum. The course is organized into four main units, each focusing on foundational concepts in algebra, functions, trigonometry, and advanced topics such as vectors and matrices. The curriculum emphasizes mathematical practices including procedural fluency, multiple representations, and mathematical reasoning.

Unit 1: Polynomial and Rational Functions

Key Topics

• Change in Tandem: Understanding how two quantities change together, often modeled by functions.

• Polynomial Functions and Rates of Change: Analysis of polynomial functions, their graphs, and how their values change.

• Polynomial Functions and Complex Zeros: Exploring the solutions of polynomial equations, including complex roots.

• Rational Functions and End Behavior: Investigating the behavior of rational functions as the input grows large or small.

• Rational Functions and Vertical Asymptotes: Identifying and interpreting vertical asymptotes in rational functions.

• Equivalent Representations of Polynomial and Rational Expressions: Converting between different forms of expressions for simplification and analysis.

• Function Model Construction and Application: Building mathematical models using polynomial and rational functions to solve real-world problems.

Key Concepts and Formulas

• Polynomial Function:

• Rational Function: , where

• End Behavior: Determined by the degrees and leading coefficients of numerator and denominator.

• Vertical Asymptote: Occurs at where and .

Example

• Find the end behavior of as .

• Solution: As , , so the function increases without bound.

Unit 2: Exponential and Logarithmic Functions

Key Topics

• Exponential Functions: Functions of the form where , , .

• Exponential Function Context and Data Modeling: Applying exponential models to real-world data, such as population growth or radioactive decay.

• Composition of Functions: Combining two or more functions, .

• Logarithmic Expressions and Functions: Understanding and manipulating logarithms, the inverse of exponentials.

• Logarithmic Function Manipulation: Applying properties of logarithms to simplify expressions and solve equations.

Key Concepts and Formulas

• Exponential Growth/Decay: , where for growth, for decay.

• Logarithm Definition:

• Properties of Logarithms:

Example

• Solve for :

• Solution:

Unit 3: Trigonometric and Polar Functions

Key Topics

• Periodic Phenomena: Modeling and analyzing repeating patterns using trigonometric functions.

• Sine, Cosine, and Tangent Functions: Definitions, properties, and applications.

• Sine and Cosine Function Values and Graphs: Calculating and graphing these functions for various inputs.

• Sinusoidal Functions and Transformations: Understanding amplitude, period, phase shift, and vertical shift.

• The Tangent Function: Properties and graphing.

• Trigonometric Equations and Inequalities: Solving equations involving trigonometric functions.

• Secant, Cosecant, and Cotangent Functions: Definitions and properties.

• Equivalent Representations of Trigonometric Functions: Using identities and alternate forms.

• Rates of Change in Polar Functions: Analyzing how polar functions change with respect to the angle.

Key Concepts and Formulas

• Sine Function:

• Period of Sine/Cosine:

• Basic Trigonometric Identities:

• Polar Coordinates: , where ,

Example

• Graph and identify amplitude, period, phase shift, and vertical shift.

• Solution: Amplitude: 2, Period: , Phase shift: right, Vertical shift: 1 up.

Unit 4: Functions Involving Parameters, Vectors, and Matrices

Key Topics

• Parametric Functions: Functions where both and are defined in terms of a third parameter, usually .

• Parametric Functions Modeling Planar Motion: Describing the motion of objects in the plane using parametric equations.

• Parametric Functions and Rates of Change: Calculating derivatives and analyzing motion.

• Parametrically Defined Circles and Lines: Expressing circles and lines using parametric equations.

• Implicitly Defined Functions: Functions defined by equations not solved for one variable in terms of another.

• Conic Sections: The study of ellipses, parabolas, and hyperbolas.

• Vectors: Quantities with both magnitude and direction, represented as .

• Vector-Valued Functions: Functions that output vectors, often used to describe motion.

• Linear Transformations and Matrices: Using matrices to represent and perform linear transformations.

• Matrices Modeling Contexts: Applying matrices to solve systems of equations and model real-world situations.

Key Concepts and Formulas

• Parametric Equations for a Circle: ,

• Vector Addition:

• Matrix Multiplication: , where

Example

• Write parametric equations for the line through with direction vector .

• Solution: ,

Mathematical Practices Emphasized

• Procedural and Symbolic Fluency: Mastery of algebraic manipulation and symbolic reasoning.

• Multiple Representations: Interpreting and connecting graphs, tables, equations, and verbal descriptions.

• Communication and Reasoning: Explaining mathematical thinking clearly and logically.

Assessment Structure

• Each unit is divided into two parts, each with multiple-choice and free-response questions.

• Progress checks are used to monitor understanding and guide further study.

Summary Table: Major Units and Topics