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
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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
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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