Section 2.5Transformation of Functions

Graphs of Common Functions

x

y

Reciprocal Function

Domain: - ,0 0,

Range: - ,0 0,

Decreasing on - ,0 0,

Odd function

and

1( )f x

x

Vertical Shifts

Vertical Shifts

Let be a function and be a positive real number.

The graph of is the graph of shifted units

vertically upward.

The graph of is the graph of shifted

f c

y f x c y f x c

y f x c y f x c

units

vertically downward.

Vertical Shifts

Example

Use the graph of f(x)=|x| to obtain g(x)=|x|-2

x

y

Horizontal Shifts

Horizontal Shifts

Let be a function and a positive real number.

The graph of is the graph of shifted

to the left units.

The graph of is the graph of shifted

to the

f c

y f x c y f x

c

y f x c y f x

right units.c

Horizontal Shifts

Example

Use the graph of f(x)=x2 to obtain g(x)=(x+1)2

x

y

Combining Horizontal and Vertical Shifts

Example

Use the graph of f(x)=x2 to obtain g(x)=(x+1)2+2

x

y

Reflections of Graphs

Refection about the -Axis

The graph of is the graph of reflected

about the -axis.

x

y f x y f x

x

Reflections about the x-axis

Reflection about the y-Axis

The graph of is the graph of reflected

about - axis.

y f x y f x

y

Example

Use the graph of f(x)=x3 to obtain the graph of g(x)= (-x)3.

x

y

Example

x

y

Use the graph of f(x)= x to graph g(x)=- x

Vertical Stretching and Shrinking

Vertically Shrinking

Vertically Stretching

x

y

x

yGraph of f(x)=x3

Graph of g(x)=3x3

This is vertical stretching – each y coordinate is multiplied by 3 to stretch the graph.

Example

Use the graph of f(x)=|x| to graph g(x)= 2|x|

x

y

Horizontal Stretching and Shrinking

Horizontal Shrinking

Horizontal Stretching

Example

x

y

Use the graph of f(x)= to obtain the

1graph of g(x)=

3

x

x

Sequences of Transformations

A function involving more than one transformation can be graphed by performing transformations in the following order:

1. Horizontal shifting

2. Stretching or shrinking

3. Reflecting

4. Vertical shifting

Summary of Transformations

A Sequence of Transformations

Move the graph to the left 3 units

Starting graph.

Stretch the graph vertically by 2.

Shift down 1 unit.

Example

x

y

1Given the graph of f(x) below, graph ( 1).

2f x

Example

x

y

Given the graph of f(x) below, graph - ( 2) 1.f x

Example

Given the graph of f(x) below, graph 2 ( ) 1.f x

(a)

(b)

(c)

(d)

x

y

Use the graph of f(x)= x to graph g(x)= -x.

The graph of g(x) will appear in which quadrant?

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

( )f x x

(a)

(b)

(c)

(d)

x

y

Write the equation of the given graph g(x). The original function was f(x) =x2

g(x)

2

2

2

2

( ) ( 4) 3

( ) ( 4) 3

( ) ( 4) 3

( ) ( 4) 3

g x x

g x x

g x x

g x x

(a)

(b)

(c)

(d)

Write the equation of the given graph g(x). The original function was f(x) =|x|

g(x)

( ) 4

( ) 4

( ) 4

( ) 4

g x x

g x x

g x x

g x x

x

y