“Teach A Level Maths”
Vol. 1: AS Core Modules
20: Stretches
© Christine Crisp
Stretches
Module C1
Module C2
Edexcel
AQA
OCR
MEI/OCR
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Stretches
We have seen that graphs can be translated.
e.g. The translation of the function y  x 3 by the
 2
vector  1  gives the function y  ( x  2) 3  1 .
 
The graph becomes
yx
3
y  ( x  2) 3  1
We will now look at other transformations.
Stretches
e.g.1 Consider the following functions:
2
y

4x
y  x2
and
For
y  x2,
For
y  4x 2 ,
x2 
x2 
y 4
y  16
In transforming from y  x 2 to y  4x 2
the y-value has been multiplied by 4
Stretches
e.g.1 Consider the following functions:
2
y

4x
y  x2
and
For
y  x2,
For
y  4x 2 ,
x2 
y 4
x  2  y  16
In transforming from y  x 2 to y  4x 2
the y-value has been multiplied by 4
Similarly, for every value of x, the y-value on
y  4x 2 is 4 times the y-value on y  x 2
yx
2

y  4x
2 is a stretch of scale factor
4 parallel to the y-axis
Stretches
The graphs of the functions are as follows:
y  4x 2
yx
(1, 4)
2
(1, 1)
y  4x 2 is a stretch of y  x 2
by scale factor 4, parallel to the y-axis
BUT, you may look at the graph and see the
transformation differently.
Stretches
y  4x
2
(1, 4)
( 2, 4 )
y  x2
y  x 2 has been squashed in the x-direction
We say there is a stretch of scale factor 1
2
parallel to the x-axis.
Stretches
y  4x 2 is a transformation of y  x 2 given by
either a stretch of scale factor 4 parallel to
the y-axis
or a stretch of scale factor 12 parallel to the xaxis
y  4x 2
y  4x 2
y  x2
4
yx
2
 12
Stretches
It is easier to see the value of the stretch in
the y direction.
To obtain y  4x 2 from y  x 2 we multiply
every value of y by 4.
The reason for the size of the 2nd stretch can
be seen more easily if we write y  4x 2 as
y  (2 x ) 2
Now, for y  x 2 ,
and for
y  (2 x ) 2 ,
x2 
x 1 
y4
y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.
To obtain y  4x 2 from y  x 2 we multiply
every value of y by 4.
The reason for the size of the 2nd stretch can
be seen more easily if we write y  4x 2 as
y  (2 x ) 2
Now, for y  x 2 ,
and for
y  (2 x ) 2 ,
x2 
x 1 
y4
y4
The x-value must be halved to give the same value
of y.
Stretches
SUMMARY
The transformation of y  x 2 to y  4x 2
yx
2

y  4x
is a stretch of scale factor
4 parallel to the y-axis
2
or
yx
2

y  (2 x )
2
is a stretch of scale factor
1 parallel to the x-axis
2
Stretches
SUMMARY
 The function
y  kf ( x ) is obtained from y  f ( x )
by a stretch of scale factor ( s.f. ) k,
parallel to the y-axis.
 The function
y  f (kx) is obtained from y  f ( x )
by a stretch of scale factor ( s.f. ) 1 ,
k
parallel to the x-axis.
Stretches
1
e.g. 2 Describe the transformation of y 
that
x
3
gives y 
.
x
Using the same axes, sketch both functions.
3
Solution: y 
x
1
can be written as y  3
x
( y  3  f ( x) )
so it is a stretch of s.f. 3, parallel to the y-axis
1
y
x
3
y
3
x
We always stretch
from an axis.
Stretches
Exercises
1. (a) Describe a transformation of y  x 2 that
2
gives y  9x .
(b) Sketch the graphs of both functions to
illustrate your answer.
Solution:
(a) A stretch of s.f. 9 parallel to the y-axis.
1
OR A stretch of s.f. 3 parallel to the x-axis.
( The 1st of these is easier, especially if we have,
2
y

8x
for example
)
(b)
y  9x 2
y  x2
Exercises
Stretches
2. The sketch below shows a function y  f ( x ) .
Copy the sketch and, using a new set of axes for
each, sketch the following, labelling the axes clearly:
(a) y  f ( 2 x )
(b) y  2 f ( x )
y  f ( x)
Describe each transformation in words.
Stretches
Solution:
y  f ( x)
(a)
(b)
y  2 f ( x)
y  f (2 x )
Stretch, s.f. 1
2
parallel to the x-axis
Stretch, s.f. 2
parallel to the y-axis
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
y  kf ( x ) is obtained from y  f ( x )
by a stretch of scale factor ( s.f. ) k,
parallel to the y-axis.
 The function
y  f (kx) is obtained from y  f ( x )
by a stretch of scale factor ( s.f. ) 1 ,
k
parallel to the x-axis.
Stretches
e.g. 1
y  4x 2 is a transformation of y  x 2 given by
either a stretch of scale factor 4 parallel to the
y-axis
or a stretch of scale factor 12 parallel to the xaxis
y  4x 2
y  4x 2
yx
2
4
yx
2
 12
Stretches
1
e.g. 2 Describe the transformation of y 
that
x
3
gives y 
.
x
Using the same axes, sketch both functions.
3
Solution: y 
x
1
can be written as y  3
x
( y  3  f ( x) )
so it is a stretch of s.f. 3, parallel to the y-axis
1
y
x
3
y
3
x
We always stretch
from an axis.