Transformations of Inverse Trigonometric Functions

Introduction

Transformations of inverse trigonometric functions follow the same principles as transformations of other functions. These transformations include translations (shifts), reflections, stretches, and compressions. Understanding these transformations helps in analyzing graphs and solving equations involving inverse trigonometric functions.

Parent Functions

The basic inverse trigonometric functions include:
  • Inverse Sine:
  • Inverse Cosine:
  • Inverse Tangent:
These functions have specific domains and ranges:
Function Domain Range
Function Domain Range sin^(-1)x [-1,1] [-(pi)/(2),(pi)/(2)] cos^(-1)x [-1,1] [0,pi] tan^(-1)x (-oo,oo) (-(pi)/(2),(pi)/(2))

General Transformation Form

A transformed inverse sine function takes the form:
where:
  • affects vertical dilation by a factor of and/or an -axis reflection if is negative.
  • affects horizontal dilation by a factor of and/or a -axis reflection if is negative.
  • affects the horizontal shift (left if , right if ).
  • affects the vertical shift (up if , down if ).

Effect of Each Parameter

1. Vertical Stretch/Compression ( )

  • If , the graph stretches vertically (elongates).
  • If , the graph compresses vertically (shrinks).
  • If , the graph reflects across the -axis.
Example:
  • Stretches the graph by a factor of .
  • Reflects across the -axis.
  • Stretches the graph by a factor of .
  • Reflects across the -axis.

2. Horizontal Stretch/Compression ( )

  • If , the graph is horizontally compressed (narrower).
  • If , the graph is horizontally stretched (wider).
  • If , the graph reflects across the -axis.
Example:
  • The graph is compressed horizontally.
  • The graph is stretched horizontally.
  • The graph is reflected across the -axis.
  • The graph is compressed horizontally.

3. Horizontal Shift ( )

  • If , the graph shifts left by units.
  • If , the graph shifts right by units.
Example:
  • The graph shifts left by unit.
  • The graph shifts right by units.
  • The graph shifts left by units.

4. Vertical Shift ( )

  • If , the graph shifts up by units.
  • If , the graph shifts down by units.
Example:
  • The graph shifts up by unit.
  • The graph shifts down by units.
  • The graph shifts up by .

Example Problem

Graph the function:

Step-by-Step Transformations

  1. Inside Parentheses: Shift right by unit.
  2. Horizontal Compression: Compress horizontally by a factor of .
  3. Vertical Stretch: Stretch vertically by a factor of .
  4. Vertical Shift: Move the graph up by .

Summary of Effects

Transformation Effect on Graph
(Vertical Stretch/Compression) Multiplies -values by
(Horizontal Stretch/Compression) Multiplies -values by
(Horizontal Shift) Moves graph left/right by
(Vertical Shift) Moves graph up/down by
Transformation Effect on Graph a (Vertical Stretch/Compression) Multiplies y-values by a b (Horizontal Stretch/Compression) Multiplies x-values by (1)/(b) c (Horizontal Shift) Moves graph left/right by c d (Vertical Shift) Moves graph up/down by d