C3 Transformations

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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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.


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


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


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


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


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.


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


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


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.


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


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


12 xey

xey 2

12 xey1

1

1

We do the sketch in 2 stages:xey xey 2

xey xey

xey 2

21


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



xy ln


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



xy ln


n:

(a) We have

xln

but for (b), xln

(a) is


the y-axis

2


1

0

)1ln( x2

xln2 1

)ln(x 1(b)

is


2 xln



xy ln


n:

(a) We have

xln

but for (b), xln

(a) is


the y-axis

2


1

0

(b) is


0

1

)1ln( x2

xln2 1

)ln(x 1

• a stretch of s.f.

2 xln


2 xln


xy ln


n:

(a) We have

xln

but for (b), xln

(a) is


the y-axis

2


1

0

(b) is


the y-axis

2


0

1

)1ln( x2

xln2 1

)ln(x 1



xy ln


n:

(a) We have

xln

but for (b), xln

(a) is


the y-axis

2


1

0

(b) is


the y-axis

2


0

1

)1ln( x2

xln2 1

)ln(x 1

2 xln


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


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


xcos

Solution:(i) a stretch of s.f. 2 parallel to the x-

axis x21cos

xy 21cos

xy cos2

stretch


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


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


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


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


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 )


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


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


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.


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.


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


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.


x

12

1

4

222

yx

The graphs look like this:

124

22

yx

122 yx


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.



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


(ii) a stretch of s.f. 5 parallel to the y-axis


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 )


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.


(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


Solution:

)2( xfy (ii))(xfy

)2( xfy

x

y

)(xfy

)(xfy

x

y

(i)

)(xfy


C3 Transformations

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