20: Stretches © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

20: Stretches20: Stretches

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

Stretches

Module C1

AQAEdexcelOCR

MEI/OCR

Module C2

"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

Stretches

3xy 1)2( 3 xy

We have seen that graphs can be translated.

e.g. The translation of the function by

the vector gives the function

.

3xy

1

21)2( 3 xy

The graph becomes

We will now look at other transformations.

Stretches

e.g.1 Consider the following functions:2xy an

d

24xy

For yxxy 2,2 4

yxxy 2,4 2For 16

In transforming from to the y-value has been multiplied by 4

2xy 24xy

Stretches

e.g.1 Consider the following functions:2xy an

d

24xy

For yxxy 2,2

yxxy 2,4 2For

Similarly, for every value of x, the y-value on

is 4 times the y-value on 24xy 2xy

22 4xyxy is a stretch of scale factor 4 parallel to the y-axis

In transforming from to the y-value has been multiplied by 4

2xy 24xy

4

16

Stretches

2xy

24xy

The graphs of the functions are as follows:

)4,1(

)1,1(

2xy is a stretch of

24xy by scale factor 4, parallel to the y-

axisBUT, you may look at the graph and see

the transformation differently.

Stretches

2xy

24xy )4,1( )4,2(

2xy has been squashed in the x-direction

We say there is a stretch of scale factor parallel to the x-axis.

21

Stretches

2xy

24xy

2xy

24xy

is a transformation of given by 24xy 2xy

either a stretch of scale factor 4 parallel to the y-axis

21or a stretch of scale factor parallel to

the x-axis

214

Stretches

It is easier to see the value of the stretch in the y direction.

2)2( xy

Now, for

4,2 yxxy 2

4,)2( 2 yxxyand for

1

The reason for the size of the 2nd stretch can be seen more easily if we write as

24xy

24xy 2xy To obtain from we multiply every value of y by 4.

The x-value must be halved to give the same value of y.

Stretches

It is easier to see the value of the stretch in the y direction.

The reason for the size of the 2nd stretch can be seen more easily if we write as

24xy 2)2( xy

24xy 2xy To obtain from we multiply every value of y by 4.

The x-value must be halved to give the same value of y.

Now, for

4,2 yxxy 2

4,)2( 2 yxxyand for

1

Stretches

22 4xyxy is a stretch of scale factor 4 parallel to the y-axis

or

22 )2( xyxy is a stretch of scale factor parallel to the x-axis21

2xy 24xy The transformation of to

SUMMARY

Stretches

SUMMARY The function

)(xkfy is obtained from )(xfy

by a stretch of scale factor ( s.f. ) k,parallel to the y-axis.

The function

)(kxfy is obtained from )(xfy

by a stretch of scale factor ( s.f. ) ,parallel to the x-axis.

k1

Stretches

We always stretch from an axis.

xy

1

xy

3

Using the same axes, sketch both functions.

so it is a stretch of s.f. 3, parallel to the y-axis

e.g. 2 Describe the transformation of

that gives .x

y1

xy

3

xy

3Solution: can be written as

xy

13

))(3( xfy

3

Stretches

(b) 2xy

29xy

Exercises1. (a) Describe a transformation of

that gives .

2xy 29xy

(b) Sketch the graphs of both functions to illustrate your answer.

Solutio

n

:

(a) A stretch of s.f. 9 parallel to the y-axis.

OR A stretch of s.f. parallel to the x-axis.

31

( The 1st of these is easier, especially if we have, for example )

28xy

Stretches

)(xfy

Copy the sketch and, using a new set of axes for each, sketch the following, labelling the axes clearly:

2. The sketch below shows a function

.

)(xfy

)2( xfy (a) )(2 xfy (b)

Describe each transformation in words.

Exercises

Stretches

)(xfy

)(2 xfy

(b)

)2( xfy

(a)

Solutio

n

:

Stretch, s.f. parallel to the x-axis

21

Stretch, s.f. parallel to the y-axis

2

Stretches

Stretches

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

Stretches

SUMMARY The function

)(xkfy is obtained from )(xfy

The function

)(kxfy is obtained from )(xfy

by a stretch of scale factor ( s.f. ) ,parallel to the x-axis.

k1

by a stretch of scale factor ( s.f. ) k,parallel to the y-axis.

Stretches

2xy

24xy

2xy

24xy

is a transformation of given by 24xy 2xy

either a stretch of scale factor 4 parallel to the y-axis

21or a stretch of scale factor parallel to

the x-axis

214

e.g. 1

Stretches

We always stretch from an axis.

xy

1

xy

3

Using the same axes, sketch both functions.

so it is a stretch of s.f. 3, parallel to the y-axis

e.g. 2 Describe the transformation of

that gives .x

y1

xy

3

xy

3Solution: can be written as

xy

13

))(3( xfy

3