c3 transformations
TRANSCRIPT
15: More 15: More TransformationsTransformations
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules
More Transformations
Module C3
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More Transformations
The translations and stretches that we met in AS can be applied to any functions.
In this presentation we will look particularly at the effect on the trig, exponential and log functions of combining transformations.
We’ll start with a reminder of some examples we’ve already met.
Try not to use a calculator when doing this topic. Graphs copied from graphical calculators look peculiar unless the scales are chosen very carefully. If you do use a calculator remember to mark coordinates of all significant points and clearly show the behaviour of the curves near the axes.
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e.g. 1 The translation of the function
by the vector gives the function
3xy
1
21)2( 3 xy
3xy 1)2( 3 xy
The graph becomes
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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
xy
3
xy
1
3
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xy cos
e.g. 3 Sketch the graph of the function xy 2cos
Solution: xyxy 2coscos
is a stretch of s.f. , parallel to the x-axis.
So,21
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xy 2cos
xy cos
e.g. 3 Sketch the graph of the function xy 2cos
Solution: xyxy 2coscos
is a stretch of s.f. , parallel to the x-axis.
So,21
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General Translations and Stretches
b
a• The function is a translation
of by)(xfy baxfy )(
Translations
Stretches
)(kxfy • The function is obtained from )(xfy by a stretch of scale factor ( s.f. ) ,parallel to the x-axis.
k1
• The function is obtained from)(xkfy )(xfy by a stretch of scale factor ( s.f. ) k,parallel to the y-axis.
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Two more Transformations
Reflection in the x-axis Every y-value changes sign when we reflect in the x-axis e.g.
So, xyxy sinsin
xy sin
xy sin
x
x
In general, a reflection in the x-axis is given by
)()( xfyxfy
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Reflection in the y-axis Every x-value changes sign when we reflect in the y-axis e.g.
So, xx eyey
xey xey x
x
In general, a reflection in the y-axis is given by
)()( xfyxfy
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SUMMARY
Reflections in the axes • Reflecting in the x-axis changes the
sign of y )()( xfyxfy
)()( xfyxfy
• Reflecting in the y-axis changes the sign of x
The examples that follow illustrate combinations of the transformations: translations, stretches and reflections.
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Combined Transformationse.g. 1 Describe the transformations of
that give the function . Hence sketch the function.
xey 12 xey
Solution:• x has been replaced by
2x:so we have a stretch of s.f.
• 1 has then been added:
xx ee 2
122 xx ee
so we have a translation of
parallel to the
x-axis
21
1
0
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The point on the y-axis . . .
We do the sketch in 2 stages:xey xey 2
xey xey
xey 2
doesn’t move with a stretch parallel to the x-axis
21
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12 xey
xey 2
12 xey1
1
1
We do the sketch in 2 stages:xey xey 2
xey xey
xey 2
21
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e.g. 2 Describe the transformations of that give(a) (b)
xy ln
1ln2 xy )1ln(2 xySolutio
n:
(a) We have
xln
but for (b), xln
(a) is
• a stretch of s.f. 2
)1ln( x2
xln2 1
)ln(x 1
2 xln
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e.g. 2 Describe the transformations of that give(a) (b)
xy ln
1ln2 xy )1ln(2 xySolutio
n:
(a) We have
xln
but for (b), xln
(a) is
• a stretch of s.f. parallel to
the y-axis
2
)1ln( x2
xln2 1
)ln(x 1
• a translation of
2 xln
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e.g. 2 Describe the transformations of that give(a) (b)
xy ln
1ln2 xy )1ln(2 xySolutio
n:
(a) We have
xln
but for (b), xln
(a) is
• a stretch of s.f. parallel to
the y-axis
2
• a translation of
1
0
)1ln( x2
xln2 1
)ln(x 1(b)
is
• a translation of
2 xln
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e.g. 2 Describe the transformations of that give(a) (b)
xy ln
1ln2 xy )1ln(2 xySolutio
n:
(a) We have
xln
but for (b), xln
(a) is
• a stretch of s.f. parallel to
the y-axis
2
• a translation of
1
0
(b) is
• a translation of
0
1
)1ln( x2
xln2 1
)ln(x 1
• a stretch of s.f.
2 xln
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2 xln
e.g. 2 Describe the transformations of that give(a) (b)
xy ln
1ln2 xy )1ln(2 xySolutio
n:
(a) We have
xln
but for (b), xln
(a) is
• a stretch of s.f. parallel to
the y-axis
2
• a translation of
1
0
(b) is
• a stretch of s.f. parallel to
the y-axis
2
• a translation of
0
1
)1ln( x2
xln2 1
)ln(x 1
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e.g. 2 Describe the transformations of that give(a) (b)
xy ln
1ln2 xy )1ln(2 xySolutio
n:
(a) We have
xln
but for (b), xln
(a) is
• a stretch of s.f. parallel to
the y-axis
2
• a translation of
1
0
(b) is
• a stretch of s.f. parallel to
the y-axis
2
• a translation of
0
1
)1ln( x2
xln2 1
)ln(x 1
2 xln
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xy ln)1ln( xy
xy ln2
xy ln
The graphs of the functions are:
(b)
1ln2 xy
(a)
)1ln(2 xy
translate stretch
stretch translate
12
1
2
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then (iii) a reflection in the x-axis
(i) a stretch of s.f. 2 parallel to the x-axis
then (ii) a translation of
2
0
e.g.3 Find the equation of the graph which is obtained from by the following transformations, sketching the graph at each stage. ( Start with ).
xy cos
20 x
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xcos
Solution:(i) a stretch of s.f. 2 parallel to the x-
axis x21cos
xy 21cos
xy cos2
stretch
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Brackets aren’t essential here but I think they make it clearer.
(ii) a translation of :
2
0 x21cos 2cos 2
1 x
2cos 21 xy
2
translate
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2cos 21 xy
(ii) a translation of :
2
0 x21cos 2cos 2
1 x
2cos 21 x 2cos 2
1 x
2cos 21 xy
2
translate reflect
x
x
(iii) a reflection in the x-axis
2cos 21 x
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Exercises1. Describe the transformations that map
the graphs of the 1st of each function given below onto the 2nd. Sketch the graphs at each stage.xey xey 2(a) to
xy ln )3ln(2 xy(b) to
xy sin xy 2sin1 (c) to
( Draw for )xsin 20 x
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xey xey 2(a) toSolution
s:
xey 2
( The order doesn’t
matter )
xey xey
Stretch s.f. 2 parallel to the y-
axis
xey xey 2
xey Reflection in the y-
axis
xey
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xy ln )3ln(2 xy(b) toSolution
s:
)3ln(2 xy
xy ln )3ln( xy Translatio
n
0
3
Stretch s.f. 2 parallel to the y-
axis
xy ln
)3ln( xy
)3ln(2 xy
)3ln( xy
( Again the order doesn’t
matter )
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xy sinSolution
s:
xy 2sin1 (c) to
xy sin xy 2sin
Translatio
n
1
0xyxy 2sin12sin Stretch s.f. parallel to the x-
axis21
Again the order doesn’t
matter.
xy sin
xy 2sin xy 2sin
xy 2sin1
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If a stretch and a translation are in the same direction we have to be very careful.
xey e.g. A stretch s.f. parallel to the y-axis on3
followed by a translation of
gives
1
0
xey xey 3 13 xeyWith the translation first, we get xey 1xey )1(3 xey
33 xey
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An important example involving stretches is the transformation of a circle into an ellipse.
122 yxe.g. Find the equation of the ellipse given by transforming the circle by
(i) a stretch of scale factor 4 parallel to the x-axis, and(ii) a stretch of scale factor 2 parallel to the y-axisMethod
Rearranging the equation of the circle to y = . . . gives a clumsy expression so we don’t do it.This means we must change the way we handle the stretch in the y direction.
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When we had , we stretched by s.f. 2 parallel to the y-axis by writing
xy ln
xyxy ln2ln
We could equally well have divided the l.h.s. by 2, so
xy
xy ln2
ln
i.e. multiplying the r.h.s. by 2.
So, to find the equation of a curve which is
stretched by 2 in the y direction, we can
replace y by
2
y
We are then treating both stretches in the same way.
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124
22
yx
(i) a stretch of scale factor 4 parallel to the x-axis, and(ii) a stretch of scale factor 2 parallel to the y-axis
Returning to the example . . .
122 yx
Solution:
Replace x by and replace y by 4
x
2
y
122 yxe.g. Find the equation of the ellipse given by transforming the circle by
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122 yx
The ellipse looks like this . . .
124
22
yx
1416
22
yx
If we want to translate the ellipse we use a similar technique
12
1
4
222
yx
e.g. to translate by replace x by and
1
2)2( x
replace y by )1( y
124
22
yx
So,
The answer is usually left in this form.
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x
12
1
4
222
yx
The graphs look like this:
124
22
yx
122 yx
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SUMMARY
we can obtain stretches of scale factor k by
When we cannot easily write equations of curves in the form
)(xfy
k
x• Replacing x by and replacing y by
k
y
we can obtain a translation of by
q
p
• Replacing x by )( px
• Replacing y by )( qx
More TransformationsExercises
422 yx1. Find the equation of the curve obtained
from with the transformations given.(i) a stretch of s.f. 3 parallel to the x-axis
and(ii) a stretch of s.f. 5 parallel to the y-axis(iii) followed by a translation of .
3
1
xy 42 (i) a stretch of s.f. 2 parallel to the x-axis
and(ii) a stretch of s.f. 5 parallel to the y-axis
2. Find the equation of the curve obtained from with the transformations given.
(iii) followed by a translation of .
2
0
2. Find the equation of the curve obtained from with the transformations given.
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(iii) followed by a translation of .
3
1
43
22
yx
(ii) a stretch of s.f. 5 parallel to the y-axis
43
22
yx
422 yx
1 (i) a stretch of s.f. 3 parallel to the x-axis
453
22
yx
45
3
3
122
yx
Solutions:
453
22
yx
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(ii) a stretch of s.f. 5 parallel to the y-axis
(iii) followed by a translation of .
2
0
xy 42 2 (i) a stretch of s.f. 2 parallel to the x-
axis
242 x
y
xy 22
xy
25
2
xy
25
2
xy
25
22
Solutions:
( or )045042 xyy
xy 22 xy 502 ( or )
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You may have to deal with a function shown only in a drawing ( with no equation given ).
If you are confident about the earlier work, try this one before you look at my solution.
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(i) (ii))(xfy )2( xfy
The diagram shows part of the curve with equation
.)(xfy
Copy the diagram twice and on each diagram sketch one of the following:
)(xfy
x
y
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Solution:
)2( xfy (ii))(xfy
)2( xfy
x
y
)(xfy
)(xfy
x
y
(i)
)(xfy
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