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Section 1.6Transformation of Functions

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Graphs of Common Functions

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−4 −3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

x

y

Reciprocal Function

( ) ( )( ) ( )

( ) ( )

Domain: - ,0 0,Range: - ,0 0,

Decreasing on - ,0 0,

Odd function

and

∞ ∪ ∞

∞ ∪ ∞

∞ ∞

1( ) f x

x=

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Vertical Shifts

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( ) ( )

( ) ( )

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.

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Vertical Shifts

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Example

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

−4 −3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

x

y

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Horizontal Shifts

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( ) ( )

( ) ( )

Horizontal Shifts

Let be a function and a positive real number.


to the left units.


to the

f c

y f x c y f x

c

y f x c y f x

• = + =

• = − =

right units.c

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Horizontal Shifts

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Example

Use the graph of f(x)=x2

to obtain g(x)=(x+1)2

−4 −3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

x

y

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Combining Horizontal and Vertical Shifts

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Example


to obtain g(x)=(x+1)2

+2

−4 −3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

x

y

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Reflections of Graphs

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( ) ( )

Refection about the -Axis

The graph of is the graph of reflectedabout the -axis.

x

y f x y f x x

= − =

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Reflections about the x-axis

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( ) ( )

Reflection about the y-Axis

The graph of is the graph of reflected

about - axis.

y f x y f x

y

= − =

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Example


to obtain the graphof g(x)= (-x)3.

−4 −3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

x

y

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Example

−4 −3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

x

y

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

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Vertical Stretching and Shrinking

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Vertically Shrinking

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Vertically Stretching

−4 −3 −2 −1 1 2 3 4

−4

−3

−2

−1

1

2

3

4

x

y

−4 −3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

x

y

Graph of f(x)=x3 Graph of

g(x)=3x3

This is vertical stretching – each y coordinate is multiplied by 3 to

stretch the graph.

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Example

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

−4 −3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

x

y

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Horizontal Stretching and Shrinking

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Horizontal Shrinking

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Horizontal Stretching

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Example

−4 −3 −2 −1 1 2 3 4 5

−4

−3

−2

−1

1

2

3

4

x

y

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

1graph of g(x)= 3

x

x

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Sequences of Transformations

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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

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Summary of Transformations

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A Sequence of Transformations

Move the graph

to the left 3 units

Starting graph.

Stretch the graphvertically by 2.

Shift down 1 unit.

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Example

−4 −3 −2 −1 1 2 3 4

−2

−1

1

2

3

4

x

y

1

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

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Example

−4 −3 −2 −1 1 2 3 4

−2

−1

1

2

3

4

x

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

+ −

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Example

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

5

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(a)

(b)(c)

(d)

4 −3 −2 −1 1 2 3 4 5 6 7 8 9 1

−3

−2

−1

1

2

3

4

x

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 IIQuadrant III

Quadrant IV

( ) f x x=

5

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(a)(b)

(c)

(d)

−1 1 2 3 4 5 6 7 8

−5

−4

−3

−2

−1

1

2

3

4

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

= + −

= − −

= + +

= − +

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(a)(b)

(c)

Write the equation of the given graph g(x).

The original function was f(x) =|x|

g(x)

( ) 4( ) 4

( ) 4

g x x g x x

g x x

= − −

= − −

= − +

5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 1

−7

−6

−5

−4

−3

−2

−1

1

2

x

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