Transformations of Functions: Precalculus Study Notes

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Functions and Graphs

Section 2.5: Transformations of Functions

This section explores the fundamental transformations applied to functions and their graphs. Understanding these transformations is essential for analyzing and graphing functions in algebra and precalculus.

Objectives

Recognize graphs of common functions.
Apply vertical and horizontal shifts to graph functions.
Use reflections to graph functions.
Perform vertical and horizontal stretching and shrinking.
Graph functions involving a sequence of transformations.

Graphs of Common Functions

Seven fundamental functions frequently encountered in algebra are:

Constant Function:
Identity Function:
Absolute Value Function:
Standard Quadratic Function:
Square Root Function:
Standard Cubic Function:
Cube Root Function:

It is essential to know the characteristics of the graphs of these functions.

Properties of Common Functions

Function	Domain	Range	Even/Odd	Increasing/Decreasing
Constant ()	All real numbers		Even	Constant everywhere
Identity ()	All real numbers	All real numbers	Odd	Increasing everywhere
Absolute Value ()	All real numbers		Even	Decreasing on , increasing on
Quadratic ()	All real numbers		Even	Decreasing on , increasing on
Square Root ()			Neither	Increasing on
Cubic ()	All real numbers	All real numbers	Odd	Increasing everywhere
Cube Root ()	All real numbers	All real numbers	Odd	Increasing everywhere

Vertical Shifts

Vertical shifts move the graph of a function up or down. If is a function and is a positive real number:

The graph of is the graph of shifted up units.
The graph of is the graph of shifted down units.

Example: Use the graph of to obtain the graph of . The graph shifts vertically up by 3 units.

Horizontal Shifts

Horizontal shifts move the graph left or right. If is a function and is a positive real number:

The graph of is the graph of shifted left units.
The graph of is the graph of shifted right units.

Example: Use the graph of to obtain the graph of . The graph shifts to the right 4 units.

Reflections of Graphs

Reflections flip the graph across an axis:

Reflection about the x-axis: The graph of is the graph of reflected about the x-axis.
Reflection about the y-axis: The graph of is the graph of reflected about the y-axis.

Example: Use the graph of to obtain the graph of . The graph reflects across the y-axis.

Vertically Stretching and Shrinking Graphs

Vertical stretching and shrinking change the steepness of the graph. If is a function and is a positive real number:

If , the graph of is vertically stretched by multiplying each y-coordinate by .
If , the graph of is vertically shrunk by multiplying each y-coordinate by .

Example: Use the graph of to obtain the graph of . The graph is vertically shrunk by a factor of 0.5.

Horizontally Stretching and Shrinking Graphs

Horizontal stretching and shrinking change the width of the graph. If is a function and is a positive real number:

If , the graph of is horizontally shrunk by dividing each x-coordinate by .
If , the graph of is horizontally stretched by dividing each x-coordinate by .

Example: Use the graph of to obtain the graph of . The graph is horizontally stretched by a factor of 2.

Graphing Using a Sequence of Transformations

To graph a function involving multiple transformations, apply each transformation in sequence. For example, to graph using the graph of :

Shift right 1 unit:
Stretch vertically by factor of 2:
Shift up 3 units:

Example: The graph of is obtained by shifting right 1 unit, stretching vertically by 2, then shifting up 3 units.