Chapter 2: Functions and Graphs

2.5 Transformations of Functions

This section explores how the graphs of functions can be manipulated through various transformations. Understanding these transformations is essential for graphing and analyzing functions in algebra.

Graphs of Common Functions

Constant Function

• Definition: The constant function is defined as , where is a constant.

• Domain: All real numbers,

• Range:

• Behavior: The function is constant (horizontal line) and is even.

Identity Function

• Definition: The identity function is .

• Domain:

• Range:

• Behavior: The function is increasing everywhere and is odd.

Absolute Value Function

• Definition:

• Domain:

• Range:

• Behavior: Decreasing on , increasing on , and even.

• Graph: V-shaped, symmetric about the y-axis.

Quadratic Function

• Definition:

• Domain:

• Range:

• Behavior: Decreasing on , increasing on , and even.

• Graph: Parabola opening upwards, vertex at the origin.

Square Root Function

• Definition:

• Domain:

• Range:

• Behavior: Increasing on , neither even nor odd.

Cubic Function

• Definition:

• Domain:

• Range:

• Behavior: Increasing everywhere, odd.

Cube Root Function

• Definition:

• Domain:

• Range:

• Behavior: Increasing everywhere, odd.

• Graph: S-shaped curve passing through the origin.

• 

Transformations of Functions

Vertical Shifts

Vertical shifts move the graph of a function up or down without changing its shape.

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

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

Horizontal Shifts

Horizontal shifts move the graph of a function left or right.

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

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

Reflections of Graphs

Reflections flip the graph of a function over a specified axis.

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

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

Vertical Stretching and Shrinking

Vertical stretching and shrinking change the steepness of a graph by multiplying the function by a constant.

• If , the graph of is vertically stretched by a factor of .

• If , the graph of is vertically shrunk by a factor of .

Horizontal Stretching and Shrinking

Horizontal stretching and shrinking affect the width of the graph by multiplying the input variable by a constant.

• If , the graph of is horizontally shrunk by a factor of .

• If , the graph of is horizontally stretched by a factor of .

Graphing Using a Sequence of Transformations

To graph a function involving multiple transformations, apply each transformation step by step in the correct order. The typical order is:

1. Horizontal shifts

2. Stretching or shrinking (vertical or horizontal)

3. Reflections

4. Vertical shifts

Example: To graph using :

• Shift left by 1 unit

• Shrink vertically by

• Reflect about the x-axis

• Shift down by 2 units

Summary Table: Types of Transformations