AP Precalculus Ultimate Guide Notes

Overview

• Study guide for AP Precalculus covering four main units:
  • Polynomial and Rational Functions
  • Exponential and Logarithmic Functions
  • Trigonometric and Polar Functions
  • Functions Involving Parameters, Vectors, and Matrices

Unit 1: Polynomial and Rational Functions

Functions and Their Properties

• Function: Maps a set of input values to output values with a unique output for each input.
• Domain & Range: Input values (x) = Domain; Output values (y) = Range.
• Increasing/Decreasing Functions: Determined by how output values change with input values.
• Concavity: Concave up (increasing rate of change), concave down (decreasing rate of change).

Polynomial Functions

• Degrees and Terms: Defined by the highest power of x; degree n has n zeros.
• Maxima/Minima: Local (relative) and global (absolute) extrema.
• Inflection Points: Where the concavity changes.
• Complex Numbers: Real and non-real zeros of polynomials.
• End Behavior: Determined by degree and leading coefficient.

Rational Functions

• Definition: Ratio of two polynomials; denominator cannot be zero.
• Asymptotes: Determine end behavior by leading terms.
• Zeros and Holes: Real zeros of numerator; holes where zero is more frequent in numerator.
• Equivalent Representations: Standard and factored forms for different analyses.

Unit 2: Exponential and Logarithmic Functions

Sequences

• Arithmetic Sequence: Constant rate of change.
• Geometric Sequence: Constant proportional change.

Exponential Functions

• Characteristics: Always increasing/decreasing, no extrema except on closed intervals.
• Transformations: Vertical and horizontal shifts and dilations.
• Base e: Natural exponential functions often use e = 2.718.

Logarithmic Functions

• Inverse of Exponential Functions: Logarithmic growth, reflections over y=x.
• Properties and Identities: Product, quotient, and power properties.

Unit 3: Trigonometric and Polar Functions

Periodic Phenomena

• Periodic Functions: Repeat at fixed intervals; characterized by period and amplitude.

Sine, Cosine, and Tangent Functions

• Angle Measurements: Degrees and radians; coterminal angles.
• Unit Circle: Helps in evaluating trigonometric functions.
• Graphing: Relationships between unit circle and graphs.

Transformations

• Sinusoidal Functions: Involving phase shifts and amplitude changes.
• Tangent Function: Periodic with transformations affecting frequency and midline.

Inverse Trigonometric Functions

• Arcsine, Arccosine, Arctangent: Inverse operations with specific domains.

Unit 4: Functions Involving Parameters, Vectors, and Matrices

Parametric Functions

• Definition: Functions involving a parameter (t) that influences x and y.
• Modeling Motion: Analyze horizontal and vertical components separately.

Vectors

• Characteristics: Direction and magnitude, represented as components.
• Operations: Addition, scalar multiplication, dot product.

Matrices

• Matrix Operations: Multiplication rules, determinants, and inverses.
• Linear Transformations: Represented by matrices, preserving vector operations.

Applications

• Conic Sections: Parametrically represented; involves ellipses, parabolas, hyperbolas.
• Utilizing Matrices: Model transitions and transformations across states.

These notes summarize the key concepts and structures of AP Precalculus, emphasizing major themes and mathematical principles discussed in the provided content.