AP Pre-Calculus Review Summary

Introduction

• Full review without Unit 4 as it's not required on the AP exam.
• Includes topic review videos minus AP style questions.
• AP style questions available in full review playlist.

Topic 1: Functions and Change

1.1 Change in Tandem

• Function: Relation where each input is mapped to exactly one output.
• Domain: Set of input values (x), independent variable.
• Range: Set of output values (y), dependent variable.
• Increasing/Decreasing Functions: Determined by input/output value relationship.
• Graph: Visual representation of input-output pairs.
• Rate of Change: Slope of the graph.
• Concavity:
  • Concave up: Rate of change increases (U-shaped).
  • Concave down: Rate of change decreases (∩-shaped).
• Zeros: Where the graph intersects x-axis (y=0).
  • Equation: y = mx + b.

1.2 & 1.3 Rate of Change

• Finding and describing rate of change (slope): y2 - y1 / x2 - x1.
• Linear Functions: Constant rate of change.
• Quadratic Functions: Changing rate of change.
• Average Rate of Change: Calculated over intervals.
• Predicting Points: Using average rate of change.

1.4 Polynomial Functions

• Polynomial Definition: No negative degrees, imaginary coefficients, or division.
• Types of Polynomials: Linear, quadratic, cubic, quartic.
• Local/Relative Max/Min: Points of highest/lowest value.
• Global/Absolute Max/Min: Greatest/least of local extremas.
• Points of Inflection: Concavity change points.

1.5 Polynomial Multiplicity & Degree

• Zeros: Where polynomial equals zero (x-intercepts, roots).
• Real vs. Imaginary Zeros: Real zeros are on graph; imaginary involve i.
• Degree: Highest power of x.
• Multiplicity: Determines interaction with x-axis.
  • Odd: Line passes through.
  • Even: Line bounces off.

1.6 End Behavior of Polynomials

• End Behavior: Graph's tendency as x approaches ±∞.
• Notation: Limit notation.
• Degree & Leading Coefficient: Determine end behavior.

Topic 2: Rational Functions & Sequences

2.1-2.5 Rational Functions and Zeros

• Rational Function: Polynomial ratio.
• Vertical & Horizontal Asymptotes: Limits of the graph.
• Finding Zeros: Set numerator to zero, solve.

2.6 Sequences

• Arithmetic Sequence: Common difference.
• Geometric Sequence: Common ratio.
• Equations: Different forms to find terms in a sequence.

2.7-2.14 Exponential and Logarithmic Functions

• Exponential Functions: Growth/decay models.
• Logarithms: Inverse of exponential functions.
• Properties of Logs: Product, quotient, power.
• Modeling with Functions: Real-world applications.

Topic 3: Trigonometric and Polar Functions

3.1-3.5 Trigonometry Basics

• Radians: Angle measure, 2π rad = 360°.
• Unit Circle: Understanding trig functions at specific angles.
• Graphs of Sin/Cos: Periodic nature of sine/cosine.

3.6-3.10 Trig Functions and Identities

• Tangent Graphs: Period and properties.
• Inverse Functions: Restricted domains.
• Identities: Reciprocal, Pythagorean, sum, and double angle.

3.11 Polar Functions

• Polar Coordinates: r, θ system.
• Graphing Polar Equations: Circles, cardioids, limacons, and roses.
• Conversion: Polar to Cartesian and vice versa.

Conclusion

• Completion of AP Pre-Calculus course content.
• Encouragement to practice and engage further with provided resources and additional content.