AP Precalculus Ultimate Guide Notes

Table of Contents

1. Polynomial and Rational Functions
2. Exponential and Logarithmic Functions
3. Trigonometric and Polar Functions
4. Functions Involving Parameters, Vectors, and Matrices

Unit 1: Polynomial and Rational Functions

Key Concepts

• Functions: Map input values (domain, independent variable) to output values (range, dependent variable).
• Increasing/Decreasing Functions: Increase or decrease in output values as input values increase.
• Graph Representation: Visual display of input-output pairs, concave up/down indicates rate of change trends.

Rates of Change

• Average Rate of Change (AROC): Slope of secant line for a function over an interval.
• Polynomial Functions: Characterized by degree, leading term, and coefficients.
• Local/Global Extrema: Maximum/minimum points where function changes direction.
• Zeros/Roots: Function's output is zero.

Rational Functions

• Rational Functions: Ratio of two polynomials; behavior determined by leading terms.
• Vertical/Horizontal Asymptotes: Determined by denominator zeros and leading term ratios.
• End Behavior: Influenced by degree and leading coefficients.

Unit 2: Exponential and Logarithmic Functions

Sequences and Functions

• Arithmetic/Geometric Sequences: Defined by common difference/ratio.
• Exponential Functions: Always increasing/decreasing, no extrema except on closed intervals.

Logarithmic Functions

• Inverse of Exponential: Represent multiplicative growth in an additive way.
• Properties: Includes Product, Quotient, Exponential, and Natural Log properties.

Unit 3: Trigonometric and Polar Functions

Periodic Phenomena

• Periodic Functions: Functions that repeat values in regular intervals.
• Sine, Cosine, Tangent: Key trigonometric functions, defined based on unit circle angles.

Sinusoidal Functions

• Transformations: Affect amplitude, period, phase shift, and vertical shift.

Inverse Trigonometric Functions

• Arcsin, Arccos, Arctan: Inverses of sine, cosine, and tangent functions.

Polar Coordinates and Graphs

• Polar vs Cartesian: Distance and angle from origin rather than x/y coordinates.
• Graphing Circles, Roses, Limacons: Defined by radius and angular relationships.

Unit 4: Functions Involving Parameters, Vectors, and Matrices

Parametric Functions

• Modeling Motion: Using parameters to describe positions over time.
• Rate of Change: Analyzing particle motion via parametric equations.

Vectors

• Components and Operations: Directional quantities with magnitude, addition, and scalar multiplication.

Matrices

• Matrix Operations: Addition, multiplication, inverses, determinants.
• Linear Transformations: Using matrices to map vectors in space.