15: More Transformations © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules.

15: More 15: More TransformationsTransformations

© Christine Crisp

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

Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules

Trig Transformations

Combined Translations

• x axis translation of –a and

y axis translation of b

a

b

y f x y f x a b( ) ( )

• x axis translation of +a and

y axis translation of b

a

b

y f x y f x a b( ) ( )

Note

Opposite sign

Note

Opposite sign


Stretches

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


Reflections

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


xy sin

e.g. 1 Sketch the graph of the function xy sin2

xyxy sin2sin is a stretch of s.f. 2, parallel to the y-axis.

Solution: We can use the fact that is a stretch of .xy sin

xy sin2



xyxy sin2sin


xy sin2

xy sin2

xy sin

is a stretch of s.f. 2, parallel to the y-axis.

The scale factor of the stretch gives the amplitude of the function.


xy cos

e.g. 2 Sketch the graph of the function xy 2cos

Solution: xyxy 2coscos

is a stretch of s.f. , parallel to the x-axis. So,

21





21

xy 2cos

xy cos

The period of is or radians.x2cos 180


Exercises

1. Give the equation of the function that is shown on the sketch below.

x

y

360

4

180

Ans: xy cos3

y

x


Solution:

A stretch of s.f. 2 parallel to the x-axis.

Sketch both functions on the same axes for the interval 2 x

2. Describe in words the transformation xyxy

21sinsin

xy sin

xy21sin

Exercises


Solution:

180180 x3. Sketch the graph of for

showing the scales clearly. What is the period of the function?

xy 3cos2

xy 3cos2The period is

1203

360

Exercises


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


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 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.

xy sin xy 2sin1(a) to

( Draw for )xsin 20 x

(b) y = cosx to y = 2cos(x – 30)


xy sinSolution

s:

xy 2sin1(a) to

xy sin xy 2sin

Translatio

n

1

0xyxy 2sin12sin Stretch s.f. parallel to the x-

axis21

xy sin

xy 2sin xy 2sin

xy 2sin1


y xcos( 30)

Vertical stretch factor

2

y x y xcos( 30) 2cos( 30)

(b) toSolution

s:

y xcos

Translation parallel to the x-

axis

30

0

-1

-2

1

2

30 60 90 120 150 180 210 240 270 300 3300X->

|̂Y

Y=cos(x)

Y=cos(x-30)

-1

-2

1

2

30 60 90 120 150 180 210 240 270 300 3300X->

|̂Y

Y=cos(x-30)

Y=2cos(x-30)


(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


In an earlier section, we met stretches.

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

33 2 xyxy e.g. is a stretch of s.f. 2, parallel to the y-axis

)()( xfkyxfy Reminder:

3xy

32xy 2

( multiplied by k ))(xf


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

k1

)()( kxfyxfy

e.g. x

yx

y2

11

is a stretch of s.f. parallel to the x-axis. 2

1

xy

1

xy

2

21

( x multiplied by k )


xy sin


xyxy sin2sin is a stretch of s.f. 2, parallel to the y-axis.


xy sin2



xyxy sin2sin


xy sin2

xy sin2

xy sin

is a stretch of s.f. 2, parallel to the y-axis.

The scale factor of the stretch gives the amplitude of the function.


xy cos




21





21

xy 2cos

xy cos

The period of is or radians.x2cos 180


Exercises

1. Give the equation of the function that is shown on the sketch below.

x

y

360

4

180

Ans: xy cos3

y

x


Solution:

A stretch of s.f. 2 parallel to the x-axis.

Sketch both functions on the same axes for the interval 2 x

2. Describe in words the transformation xyxy

21sinsin

xy sin

xy21sin

Exercises


Solution:

180180 x3. Sketch the graph of for

showing the scales clearly. What is the period of the function?

xy 3cos2

xy 3cos2The period is

1203

360

Exercises

15: More Transformations © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules.

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Transcript of 15: More Transformations © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules.