Transformation of FunctionsSec. 1.7

Objective

You will learn how to identify and graph transformations.

Important Vocabulary

Vertical Shift – the parent graph shifts up or down

Horizontal Shift – the parent graph shifts left or right

Nonrigid Transformations – the size of the parent graph changes but the shape remains the same

Shifting Graphs

Vertical shift c units upward:

Vertical shift c units downward:

Horizontal shift c units to the right:

Horizontal shift c units to the left:

Example

Write the equation for the function resulting from a vertical shift of 3 units downward and a horizontal shift of 2 units to the right of the graph of f(x).

Example

Use the graph of to sketch the graph of each function. a.)

Example

Use the graph of to sketch the graph of each function. b.)

A family of functions is…

A set of functions (or graphs) that have the same shape but are at a different location in the plane.

Reflecting Graphs

A reflection in the x-axis is a type of transformation of the graph represented by

. A reflection in the y-axis is a type of transformation of the graph represented by

.

Example

Let . Describe the graph of in terms of f.

Example

The function g(x) shown in the figure is a transformation of the graph of . Find an equation for the function g(x).

Example

Compare the graphs of each function with the graph of

a.)

b.)

The graph of g(x) is a reflection of f(x) across the x-axis.

The graph of h(x) is a reflection of f(x) across the x-axis, and it is shifted left two units.

Rigid Transformations

A rigid transformation is a transformation that does not alter the size or shape of a function.

Rigid transformations change only the position of the graph in the xy-plane.

Types of rigid transformations: Horizontal shifts Vertical shifts Reflections

Nonrigid Transformations

Types of nonrigid transformations: Vertical stretch Vertical shrink Horizontal stretch Horizontal shrink

Nonrigid Transformations

A vertical stretch is represented by where c>1.

A vertical shrink is represented by where 0<c<1.

A horizontal shrink is represented by where c>1.

A horizontal stretch is represented by where 0<c<1.

Example

Compare the graph of each function with the graph of

a.)

b.) The graph of g(x) is a vertical stretch of f(x).

The graph of h(x) is a vertical shrink of f(x).

Example

Compare the graph of each function with the graph of

a.)

The graph of g(x) is a horizontal shrink of f(x).

Compare

Example

Compare the graph of each function with the graph of

b.)

The graph of h(x) is a horizontal stretch of f(x).

Compare

Practice Problems

Sec. 1.7, page 72 – 75

# 21 – 45 every other odd