Copyright © 2011 Pearson Education, Inc. Slide 2.3-1

2.3 Vertical Stretching

.1 units, stretched

)( ofgraph General

cc

xfy

. 2.3 and , 4.2

, 3.4, ofgraph The

43

21

xyxy

xyxy

Vertical Stretching of the Graph of a Function

If a point lies on the graph of then the point lies on the graph of If then the graph of is a vertical stretching of the graph of by applying a factor of c.

( ).y cf x( ),y f x ( , )x cy( , )x y

1,c ( )y cf x( )y f x

Copyright © 2011 Pearson Education, Inc. Slide 2.3-2

2.3 Vertical Shrinking

.10 units, shrunk

)( ofgraph General

cc

xfy

.3

4

3

3

3

2

3

1

3. and ,5.

,1., ofgraph The

xyxy

xyxy

Vertical Shrinking of the Graph of a Function

If a point lies on the graph of then the point lies on the graph of If then the graph of is a vertical shrinking of the graph of by applying a factor of c.

( ).y cf x( ),y f x ( , )x cy( , )x y

0 1,c ( )y cf x( )y f x

Copyright © 2011 Pearson Education, Inc. Slide 2.3-3

2.3 Horizontal Stretching and Shrinking

( )y f cx

Horizontal Stretching and Shrinking of the Graph of a Function

If a point lies on the graph of then the point lies on the graph of

(a) If then the graph of is a horizontal stretching of the graph of

(b) If then the graph of is a horizontal shrinking of the graph of

( ).y f cx( ),y f x( , )x y

0 1,c ( )y f cx( ).y f x

( / , )x c y

( ).y f x1,c

Copyright © 2011 Pearson Education, Inc. Slide 2.3-4

2.3 Reflecting Across an Axis

Reflecting the Graph of a Function Across an Axis

For a function defined by the following are true.(a) the graph of is a reflection of the graph of f across the x-axis.(b) the graph of is a reflection of the graph of f across the y-axis.

)(xfy ),(xfy

)( xfy

Copyright © 2011 Pearson Education, Inc. Slide 2.3-5

2.3 Example of Reflection

Given the graph of sketch the graph of

(a) (b)

Solution

(a) (b)

),(xfy )(xfy )( xfy

).,( is so

,graph on the is ),(point If

ba

ba

If point ( , ) is on the graph, so is ( , ).

a ba b

Copyright © 2011 Pearson Education, Inc. Slide 2.3-6

2.3 Combining Transformations of Graphs

ExampleDescribe how the graph of can be obtained by transforming the graph of Sketch its graph.

Solution

Since the basic graph is the vertex of the parabola is shifted right 4 units. Since the coefficient of is –3, the graph is stretched vertically by a factor of 3 and then reflected across the x-axis. The constant +5 indicates the vertex shifts up 5 units.

5)4(3 2 xy.2xy

,2xy 2)4( x2)4(3 x

2) 53( 4xy

shift 4 units right

shift 5 units up

vertical stretch by a factor of

3

reflect across the x-axis

Copyright © 2011 Pearson Education, Inc. Slide 2.3-7

Graphs:

5)4(3 2 xy

2( 4)y x 23( 4)y x

23( 4)y x

Copyright © 2011 Pearson Education, Inc. Slide 2.3-8

2.3 Caution in Translations of Graphs

• The order in which transformations are made can be important. Changing the order of a stretch and shift can result in a different equation and graph.

– For example, the graph of is obtained by first stretching the graph of by a factor of 2, and then translating 3 units upward.

– The graph of is obtained by first translating horizontally 3 units to the left, and then stretching by a factor of 2.

32 xyxy

32 xy

Copyright © 2011 Pearson Education, Inc. Slide 2.3-9

2.3 Transformations on a Calculator-Generated Graph

Example

The figures show two views of the graph and another graph illustrating a combination of transformations. Find the equation of the transformed graph.

SolutionThe first view indicates the lowest point is (3,–2), a shift 3 units to the right and 2 units down. The second view shows the point (4,1) on the graph of the transformation. Thus, the slope of the ray is

Thus, the equation of the transformed graph is

xy

First View Second View

.31

3

43

12

m

.233 xy