AP Precalculus Study Guide

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Unit One

Functions

• Concave up: Function is changing at an Increasing Rate
• Concave down: Function is changing at a Decreasing Rate
• Inflection Point: The point where the function’s concavity changes
• End Behaviors: Usually shown in limit notation “ ”

• Example: f() =

• Right/Left Side Limits:

• As the function approaches from the right:
• As the function approaches from the left:
• Typical Format:

• Average Rate of Change Formula:
• Secant Line: Line formed between two points using the Rate of Change Formula
• Asymptotes (Rational Equation Form):

For Reference:

• Vertical Asymptote 

• Set Denominator to Zero

• Horizontal Asymptote (y =)

• n<m ⇒  
• n=m ⇒  
• n>m ⇒ No Slant Asymptote

• Slant Asymptote

• Solve using Long division (Do not include remainder)

• Change Function Patterns

• Linear

• The rate of change is constant

• Quadratic

• Constant 2nd output value differences over equal length inputs

• Polynomial (Degree n)

• nth differences of output values are constant over equal-length input intervals

• Exponential

• Outputs are proportional over equal-length input values

• Logarithmic

• Input values are proportional over equal-length output values

Unit Two

Sequences

• Arithmetic Sequences:

• Changes linearly
• Format:

• Geometric Sequences:

• Changes exponentially
• Format:
                  
                      r = ratio

Functions

• Exponential Functions:

• Growth
• Decay
• Properties:

• 
• 
• 
• (
• =1
• 
• 

• Logarithmic Functions:

• Remember:  
• Properties:

• Euler’s Constant: “ ”

• Properties:

•   &  
• 

• Inverse Functions

• When the function’s inputs and outputs are interchanged

• Example: regular (x,y) ⇒ (y,x) inversed

• Properties:

•   &  

Unit Three

Sequences

• Inverse Trigonometric Function Properties

• Trigonometric Graphs:

Trigonometric Functions. (n.d.). Graph of Trigonometric Functions. Study.com Retrieved from https://study.com/cimages/multimages/16/graphtrigonometricfunctions2.jpg

• Unit Circle: Make sure you memorize this!

• Trigonometric Function Transformations

• Formula:
• For  and and its inverses

• A - Amplitude
• B - Period

• Solve for Period with

• C - Phase Shift (Note the negative)
• D - Midline

• For  and its inverse

• A - Amplitude
• B - Period

• Solve for Period with

• C - Phase Shift (Note the negative)
• D - Midline

• Pythagorean Identities

• Other Identities

• Polar Graph

• Formulas:

• )
• )

• Properties:

• Coordinates and Form Conversion

• Rectangular Coordinate (x,y) ⇒ Polar Coordinate (r,θ)

• 
•  reference angle
  (use quadrant and reference angle to find θ)

• Polar Coordinate (r,θ) ⇒ Rectangular Coordinate (x,y)

• Polar Form ⇒ Complex Number

•  ⇒

• Rose Curve Functions

• Formulas:

• Properties:

• If  is even, then the graph will have  pedals
• If  is odd, then the graph will have  pedals