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

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


Chapter 2: Analysis of Graphs of Functions

2.3 Stretching, Shrinking, and Reflecting Graphs

Goals: Recognize difference between x-axis & y-axis reflections Apply vertical stretches (compressions) Analyze functions and determine domains & ranges

graphically and analytically. Apply series of transformations to a parent function to

produce a graphical representation of a function efficiently. Synthesize a functions equation from graphical

representations


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


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


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

This topic is very difficult to recognize when looking at a graph and trying to determine the equation of the function because a horizontal stretch looks very similar to a vertical

compression. And a horizontal compression looks like a vertical stretch (horizontal stretching and shrinking will be dealt with in greater detail when we discuss

trigonometric (periodic) functions.


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

If (3,4) is a point on f(x), where does that point go in –f(x)?

If (3,4) is a point on f(x), where does that point go in f(-x)?


2.3 Example of Reflection

Given the graph of sketch the graph of(a) (b)

Solution(a) (b)

),(xfy

)(xfy )( xfy


2.3 Combining Transformations of Graphs

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

What is the parent function?

Solution:What is the vertex of the original parent function?What is the new vertex after transformations?What was the domain and range of parent function?What is the new domain and range?

2( ) 3( 4) 5f x x

2) 53( 4xy


• Why is this set of steps important? If we take y=-x2+2 and graph it and apply the vertical translation first and then reflect it, the process will provide us a graph that is not consistent with what our technology provides. If we do some algebra we can rewrite it as y=-(x2-2) thus we can graph the parent function, slide it down two units and then reflect over x axis. However some people do not like doing the preliminary algebra. SO what many people do is a vertical shift up 2 units and then reflect over the x-axis which provides a completely different (incorrect) graph. HOWEVER, if you follow the steps below and basically save all vertical shifts for last you will never make this mistake.

(L/R SHIFT)

(U/D SHIFT)


5)4(3 2 xy2( 4)y x 23( 4)y x

Determine domains and ranges after quickly sketching the graphs, can these be found without graphing?

Parent functionCritical points?

Domain:Range:

Domain:Range:

Domain:Range:

Domain:Range:


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


2.3 Transformations on a Calculator-Generated Graph

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

Solution: In order to find the scalar that does the stretching we must find the slope of one of the rays of the curve.

xy

First View Second View


Now lets complete transformations on an abstract function f(x)


Extend your thinkingReinforce learned concepts

• www.faymathematics.pbworks.com

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

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