Copyright © 2007 Pearson Education, Inc. Slide 2-1 Chapter 2: Analysis of Graphs of Functions 2.1...

Copyright © 2007 Pearson Education, Inc. Slide 2-1

Chapter 2: Analysis of Graphs of Functions

2.1 Graphs of Basic Functions and Relations; Symmetry

2.2 Vertical and Horizontal Shifts of Graphs

2.3 Stretching, Shrinking, and Reflecting Graphs

2.4 Absolute Value Functions: Graphs, Equations, Inequalities, and Applications

2.5 Piecewise-Defined Functions

2.6 Operations and Composition


2.3 Stretching, Shrinking, and Reflecting Graphs

Vertical Stretching of the Graph of a Function

If c > 1, the graph of is obtained by vertically stretching the graph of by a factor of c. In general, the larger the value of c, the greater the stretch.

)(xfcy )(xfy

.1 units, stretched

)( ofgraph General

cc

xfy

. 2.3 and , 4.2

, 3.4, ofgraph The

43

21

xyxy

xyxy


2.3 Vertical Shrinking

Vertical Shrinking of the Graph of a Function

If the graph of is obtained by vertically shrinking the graph of by a factor of c. In general, the smaller the value of c, the greater the shrink.

,10 c )(xfcy )(xfy

.10 units, shrunk

)( ofgraph General

cc

xfy

.3

4

3

3

3

2

3

1

3. and ,5.

,1., ofgraph The

xyxy

xyxy


2.3 Horizontal Stretching

Horizontal stretching of a graph of a function

If 0 < c < 1, then the graph of y = f (cx ) is a horizontal stretching of the

graph of y = f ( x ). In general, smaller the value of c, greater the stretch.

Horizontal stretching changes x - intercepts but not the y - intercept


2.3 Horizontal Shrinking

Horizontal shrinking of a graph of a function

If c > 1, then the graph of y = f (cx ) is a horizontal shrinking of the

graph of y = f ( x ). In general, larger the value of c, greater the shrink

Horizontal shrinking changes x - intercepts but not the y - intercept


2.3 Reflecting Across an Axis

Reflecting the Graph of a Function Across an Axis

For a function(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


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


2.3 Reflection with the Graphing Calculator

).(

and ,

,126Set

13

12

2

1

xyy

yy

xxy

. and ofgraph thehave We 21 yy

. and ofgraph thehave We 31 yy


2.3 Combining Transformations of Graphs

Example

Describe 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( x 2)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


Graphs:

5)4(3 2 xy

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

23( 4)y x


2.3 Caution in Translations of Graphs

• The order in which transformations are made is important. If they are made in a different order, a different equation can result.

– 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


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

Copyright © 2007 Pearson Education, Inc. Slide 2-1 Chapter 2: Analysis of Graphs of Functions 2.1...

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