CHAPTER 36... · 584 CHAPTER 36 Transformations of functions So the transformation that maps the...

582

CHAPTER 36 Transformations of functions

36.1 Function notationA function is a rule that changes a number.

A function can be thought of as a machine.A number called the input is put into the machine.The machine then changes the number and gives the result (the output).

Let the letter f stand for the rule ‘double the input and then add 3’

If x is the input, the output will be 2x � 3

This rule or function can be written as f(x) � 2x � 3

f(4), read as ‘f of 4’, means work out the output when the input is 4

This is written as f(4) � 2 � 4 � 3

� 11

In general terms, if x is the input then f(x) is the output

f(x) is an example of function notation and is read as ‘f of x’.Any letter can be used for a function although f, g and h are the letters most commonly used.

f(x) � 3x � 5 and g(x) � x2 � 7 Find the value of a f(2) b g(�3) c f(0) � g(0)

Solution 1a f(x) � 3x � 5

f(2) � 3 � 2 � 5

� 1

b g(x) � x2 � 7

g(�3) � (�3)2 � 7

� 16

c f(x) � g(x) � (3x � 5) � (x2 � 7)

f(0) � g(0) � (3 � 0 � 5) � (02 � 7)

� 2

Substitute 0 for x

Substitute �3 for x

Substitute 2 for x

Example 1

f f(x)x

f 2 � 4 � 3 � 114

f 2 x � 3x

function OutputInput

36C H A P T E R

Transformations offunctions

583

36.2 Applying vertical translations CHAPTER 36

f(x) � 2x � 1Find an expression for a f(x) � 4 b 3f(x) c f(�x)

Solution 2a f(x) � 4 � 2x � 1 � 4

� 2x � 5

b 3f(x) � 3(2x � 1)

c f(x) � 2x � 1

f(�x) � 2 � (�x) � 1

� 1 � 2x

Exercise 36A

1 f(x) � 5x � 2 Find the value of

a f(2) b f(6) c f(�3) d f(0) e f(�12�)

2 g(x) � x2 � 3 Find the value of

a g(4) b g(0) c g(�1) d g(�14�) e g(��

12�)

3 f(x) � x2 Find an expression for

a f(x) � 1 b f(x) � 3 c 2f(x) d �14�f(x) e 3f(x) � 1

4 f(x) � x2 Find an expression for

a f(x � 1) b f(x � 3) c f(2x) d f(�14�x) e f(3x � 1)

5 h(x) � 3x � 2 Find an expression for

a h(x) � 3 b h(x � 3) c h(2x) d h(�x) e �h(x)

36.2 Applying vertical translationsThe graphs of y � x, y � x � 1 and y � x � 2 are shown.

The three equations are all in the form y � x � c so all the lines have a gradient of 1

Therefore the lines are parallel.

The line y � x can be moved verticallyonto each of the other lines.

If the line y � x is moved up by 1 unitthen it becomes the line y � x � 1

You can confirm this using tracing paper:

● trace the graph of y � x● move the tracing paper one unit vertically upwards

● the line drawn on the tracing paper will be directly over the graph of y � x � 1

�4

�3

�2

�1

1

O

2

3

4

y

y � x � 2

y � x � 1

�5�6 �4 �3 �2 �1 1 2 3 4 5 6 x

y � x

Replace x by �x

Replace f(x) with 2x � 1

Simplify.

Replace f(x) with 2x � 1

Example 2

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So the transformation that maps the graph of y � x onto the graph of y � x � 1 is a translation of1 unit vertically in the positive y direction.

This can be written as a translation with vector � �.Tracing paper can be used to confirm that if the graph of y � x is translated 2 units vertically in the

negative y direction or by � �, then its equation becomes y � x � 2

a The graph of y � x is translated by 5 units vertically in the positive y direction.Write down the equation of the new graph.

b The graph of y � x is translated by 3 units vertically in the negative y direction.Write down the equation of the new graph.

c Describe the transformation that will map the graph of y � x onto the graph of y � x � 8

Solution 3a y � x � 5

b y � x � 3

c Translation of 8 units vertically in the positive y direction

or

Translation of � �

The graphs of y � x2, y � x2 � 6 and y � x2 � 2 are shown opposite.

Function notation can be used when drawing graphs.

If f(x) � x2 then y � x2

can be written as y � f(x)

y � x2 � 6can be written as y � f(x) � 6

y � x2 � 2can be written as y � f(x) � 2

You can use tracing paper to confirm that the transformation that maps the graph of y � f(x) onto

the graph of y � f(x) � 6 is a translation of � � or 6 units vertically in the positive y direction

and

the transformation that maps the graph of y � f(x) onto the graph of y � f(x) � 2 is a translation of

� � or 2 units vertically in the negative y direction.

For any function, f, the transformation which maps the graph of y � f(x) onto the graph of y � f(x) � a

is a translation of � �.If a � 0 this is a translation in the positive y direction.

If a � 0 this is a translation in the negative y direction.

0a

0�2

06

y � x2 � 2

y � x2 � 6

y � x2

5

O

10

15

20

25

y

�6 �4 �2 2 4 6 x

08

The graph is translated 3 unitsvertically in the negative y direction.

The graph is translated 5 unitsvertically in the positive y direction.

Example 3

0�2

01

585

36.2 Applying vertical translations CHAPTER 36

The graph of y � x2 � 2x is shown in black.

Trace the graph y � x2 � 2x and use the tracing to help you write down the equation for each of the other two graphs.

Solution 4a y � x2 � 2x � 8

b y � x2 � 2x � 5

It is possible to transform a graph without knowing its equation.

The graph of y � f(x) is shown in black. Sketch on the same axes the graph of

a y � f(x) � 3

b y � f(x) � 2

Solution 5a The equation of y � f(x) � 3 is of the

form y � f(x) � a with a � 3 so to draw the graph of y � f(x) � 3, translate the graph of y � f(x) by 3 unitsvertically in the positive y direction

(or by � �). The graph of y � f(x) � 3 is drawn in red.

b Similarly the graph of y � f(x) � 2 is a translation of y � f(x) by 2 units in the negative

y direction (or by � �).The graph of y � f(x) � 2 is drawn in blue.

0�2

03

(b)

(a)

O

15

20

y

�5 5 x

10

5

Example 5

Move the tracing paper down 5 units y � x2 � 2x has beentranslated 5 units vertically in the negative y direction.

Move the tracing paper up 8 units y � x2 � 2x has beentranslated 8 units vertically in the positive y direction.

(b)

(a)

5

�5

O

15

20

y

�5 5 x

10

Example 4

586


Exercise 36B

1 The equation of the graph drawn in black is given.Write down the equation of each of the other two graphs. You may find it useful to use tracingpaper and trace the original graph each time.

i ii

2 a The graph of y � x2 is translated by 12 units vertically in the positive y direction.Write down the equation of the new graph.

b The graph of y � x2 is translated by 8 units vertically in the negative y direction.Write down the equation of the new graph.

c Describe the transformation that will map the graph of y � x2 onto the graph of y � x2 � 7

2

�2

�4

�6

�8

O

6

8

y

�2 2 x

42

�2

�4

O

6

y

�2�4 2 x

4

4 The graph of y � g(x) is shown.Copy and sketch on the same axes thegraph of

a y � g(x) � 7 b y � g(x) � 4

3 The graph of y � f(x) where f(x) � x2 � 2xis shown.Copy and sketch on the same axes the graph of

a y � f(x) � 1 b y � f(x) � 2

(b)

(a)

5

�5

O

15

y

�2�4 2 4 x

10

y � x 2

(b)

(a) 5

�5

O

y

�4 �2 642 x

y � x

5 The graph of y � f(x) is shown.On the resoure sheet sketch on the same axesthe graph of

a y � f(x) � 2

b y � f(x) � 1

36.3 Applying horizontal translationsGraphs can be translated horizontally.

The graphs of y � f(x), y � f(x � 3) and y � f(x � 2) where f(x) � x2 are shown below:

● trace the graph of y � f(x)

● move the tracing paper 3 units to the left which is the negative x direction

● the curve drawn on the tracing paper is directly over the graph of y � f(x � 3)

The transformation that maps the graph of y � f(x) onto the graph of y � f(x � 3) is a translation of3 units horizontally in the negative x direction.

This can be written as a translation of � �.Tracing paper can be used to confirm that if the graph of y � f(x) is translated 2 units horizontally in

the positive x direction or by � �, then it will coincide with the graph of y � f(x � 2).

For any function, f, the transformation which maps the graph of y � f(x) onto the graph of

y � f(x � a) is a translation of � �.If a � 0 this is a translation in the negative x direction.

If a � 0 this is a translation in the positive x direction.

�a0

20

�30

O

y

�8 �2�4�6 2 4 6 8 x

25

20

15

10

5y � f(x � 3) y � f(x � 2)

y � f(x)

1

�1

�2

�3

�4

O

3

y

�2 2 x

2

587

36.3 Applying horizontal translations CHAPTER 36

588


Here is a sketch of the curve with equation y � f(x).

The vertex, A, of the curve is (1, 2).

Write down the coordinates of the vertex of each of the curves with these equations.

a y � f(x � 3)

b y � f(x � 1)

Solution 6a The vertex of the curve y � f(x � 3) will have

coordinates (1 � 3, 2) � (�2, 2)

b The vertex of the curve y � f(x � 1) will have coordinates (1 � 1, 2) � (2, 2)

Transformations can be combined.

The graph of y � f(x) is shown.Sketch the graph of y � f(x � 3) � 2

�2

O

y

�2�4 2 x

2

4

Example 7

�4

�3

�2

�1O

y

�2 2 4 x

2

1

y � f(x � 1)

�4

�3

�2

�1O

y

�2�4 2 4 x

2

1

y � f(x � 3)

�4

�3

�2

�1O

y

�2 2 4 x

4

3

2

1

A(1, 2)

Example 6

The graph of y � f(x � 3) is a translation of y � f(x)by 3 units horizontally in the negative x direction.The vertex will move 3 units horizontally in thenegative x direction. So its y-coordinate will remainthe same but its x-coordinate will decrease by 3

The graph of y � f(x � 1) is a translation of y � f(x)by 1 unit horizontally in the positive x direction.The vertex will move 1 unit horizontally in thepositive x direction. So its y-coordinate will remainthe same but its x-coordinate will increase by 1

Solution 7Step 1Sketch the graph of y � f(x � 3).y � f(x � 3) is a horizontal translation of y � f(x) by 3 units horizontally in the negative x direction.

y � f(x � 3) is drawn in blue.

Step 2Sketch the graph of y � f(x � 3) � 2y � f(x � 3) � 2 is a vertical translation of y � f(x � 3) by 2 units vertically in the positive y direction

y � f(x � 3) � 2 is drawn in red.

Exercise 36C

1 The equation of the graph drawn in black is given each time.Write down the equation of each of the other two graphs.

i

ii

2 The graph of y � f(x) where f(x) � x2 � 2x is shown.On the resource sheet sketch on the same axes the graph of

a y � f(x � 4)

b y � f(x � 3)

O

y

�4�8 �2�6 4 4 82 x

2

�2

�4

4

6

O

y

�2�4�6�8 2 4 6 8 x

10

20

25

5

15 y � x 2(b)

(a)

O

y

�2�4�6 2 4 6 x

4

3

2

1

�1

�2

�3

�4

(a)

(b)

y � x

589

36.3 Applying horizontal translations CHAPTER 36

O

y

�2�4�6 2 x

2

4

6

y � f(x � 3)

y � f(x � 3) � 2

y � f(x)

3 The graph of y � f(x) is shown.On the resource sheet sketch on the same axes the graph of

a y � f(x � 2)

b y � f(x � 4)

4 This is a sketch of the curve with equation y � f(x).The vertex of the curve is (0, 2). Write down the coordinates of the vertex for each of the curves with these equations.

a y � f(x � 3)

b y � f(x � 1)

c y � f(x) � 2

d y � f(x) � 3

5 The graph of y � f(x) is shown.On the resource sheet sketch the graph of

a y � f(x � 2) � 1

b y � f(x � 2) � 1

36.4 Applying reflections

The diagrams show the graphs of y � f(x) in black and y � �f(x) in red for f(x) � x , f(x) � x2 and f(x) � (x � 1)3

In each case the graph of y � �f(x) is the reflection in the x-axis of the graph of y � f(x).

When a graph is reflected in the x-axis the x-coordinate of every point remains the same whilst thesign of the y-coordinate changes.

For any function, f, the transformation which maps the graph of y � f(x) onto the graph of y � �f(x)is a reflection in the x-axis.

y

2�2 xO

4

�1

�2

�3

�4

3

2

1

y � �f(x)

y � f(x)y

2�2 xO

4

�1

�2

�3

�4

3

2

1

y � �f(x)

y � f(x)

y

42�4 �2 O

4

�1

�2

�3

�4

3

2

1y � �f(x) y � f(

O

y

�2 2 4 x

2

�2

�4

4

6

8

O

y

�2 2

(0, 2)

x

1

�2

�4

�1

�3

2

4

3

O

y

�4�8 �2�6 42 x

2

�2

�4

4

6

590


591

36.4 Applying reflections CHAPTER 36

The graph of y � f(x) where f(x) � x2 � 4 is shown in black.

a Sketch the graph of y � �f(x).

b Write down the equation of the graph of y � �f(x).

Solution 8a The graph of y � �f(x) is a reflection in the

x-axis of the graph of y � f(x).The graph of y � �f(x) is drawn in red.

b f(x) � (x2 � 4)

so �f(x) � �(x2 � 4)

� 4 � x2

The equation of graph of y � �f(x) is y � 4 � x2

The diagrams below show the graphs of y � f(x) and y � f(�x) for f(x) � x, f(x) � (x � 1)2

and f(x) � (x � 1)3

In each case the graph of y � f(�x) is the reflection in the y-axis of the graph of y � f(x).

When a graph is reflected in the y-axis the y-coordinate of every point remains the same whilstthe sign of the x-coordinate changes.

For any function, f, the transformation which maps the graph of y � f(x) onto the graph of y � f(�x) is a reflection in the y-axis.

The diagram shows the graph of y � f(x) where f(x) � x2 � 4x � 3 drawn in black.

a Sketch the graph of y � f(�x).

b Write down the equation of the graph of y � f(�x).

Solution 9a The graph of y � f(�x) is a reflection in the y-axis of

the graph of y � f(x). The graph of y � f(�x) is drawn in blue.

b f(x) � x2 � 4x � 3

so f(�x) � (�x)2 � 4(�x) � 3

� x2 � 4x � 3

The equation of the graph of y � f(�x) is y � x2 � 4x � 3

y

y � f(x)

42�4 �2 xO

2

4

Example 9

y

2�2 xO

1

�1

�2

�3

2

4

5

3

y � f(�x)

y � f(x)

y

42�4 �2 xO

2

4

6

y � f(�x) y � f(x)

y

42�4 �2 xO

4

�1

�2

�3

�4

3

2

1y � f(�x) y � f(x)

Example 8y

2�2 xO

4

6

�2

�4

�6

2

y � f(x)

Here is a sketch of the curve with equation y � f(x).The vertex of the curve is (2, 1).Write down the coordinates of the vertex for each of the curves with these equations.

a y � �f(x)

b y � f(�x)

Solution 10a The vertex of the curve y � �f(x)

will have coordinates (2, �1)

b The vertex of the curve y � f(�x) will have coordinates (�2, 1)

Exercise 36D

1 Use the resource sheet and for each graph draw on the same axes the graph of i y � �f(x) ii y � f(�x)

a b c

2 The graph of y � 2x � 3 is reflected in the x-axis. Write down the equation of the new graph.

3 The graph of y � x(x � 2) is reflected in the y-axis. Write down the equation of the new graph.

4 The graph of y � f(x) has a vertex at (3, �1). Write down the coordinates of the vertex for eachof the curves with these equations a y � �f(x) b y � f(�x)

y

2�2 xO

�2

�1

�3

2

3

1y � f(x)

y

42�4 �2 xO

4

�2

�4

�1

�3

2

3

1

y � f(x)

y

42�4 �2 xO

4

�2

�4

�1

�3

2

3

1y � f(x)

y

5�5 xO

4

�2

�4

2

y � f(�x)

y � �f(x)

y � f(x)

The graph of y � �f(x) is a reflection in the x-axis of thegraph of y � f(x). So the x-coordinate of the vertex willremain the same but the y-coordinate will change sign.

y

y � f(x)

42 xO

2(2, 1)

1

�1

�2

�3

Example 10

592


The graph of y � f(�x) is a reflection in the y-axis of thegraph of y � f(x). So the y-coordinate of the vertex willremain the same but the x-coordinate will change sign.

593

36.5 Applying stretches CHAPTER 36

36.5 Applying stretchesThe two diagrams show the graphs of y � f(x), y � 2f(x) and y � �

12�f(x) for the functions f(x) � x

and f(x) � x2

For any function, f, the transformation which maps the graph of y � f(x) onto the graph of y � af(x)for a � 0, is a stretch of scale factor a parallel to the y-axis.

The graph y � f(x) where f(x) � x2 � 4x � 1 is drawn in black.

a Sketch the graph of y � 2f(x).

b Sketch the graph of y � �13�f(x).

y

2 4�2 xO

4

�2

�4

�6

2

Example 11

y

2 4 6�2�4�6 xO

2

4

6

8

10

12

y � x 2

x2

2y �

y � 2 x2

Points on the graph of y � x are mapped ontocorresponding points on the graph of y � 2x.For example (1, 1) → (1, 2), (2, 2) → (2, 4) and (�1, �1) → (�1, �2).

Similarly points on the graph of y � x2 are mapped ontocorresponding points on the graph of y � 2x2

For example (1, 1) → (1, 2), (2, 4) → (2, 8) and (�1, 1) → (�1, 2).

In all cases the x-coordinate has remained unchangedand the y-coordinate has been multiplied by 2

This transformation is called a stretch. The scale factorof the stretch is 2 and the direction of the stretch isparallel to the y-axis.

Points on the graph of y � x are mapped ontocorresponding points on the graph of y � �

12�x.

For example (1, 1) → (1, �12�), (2, 2) → (2, 1) and

(�1, �1) → (�1, ��12�).

Points on the graph of y � x2 are mapped ontocorresponding points on the graph of y � �

12�x2

For example (1, 1) → (1, �12�), (2, 4) → (2, 2) and

(�1, 1) → (�1, �12�).

In all cases the x-coordinate has again remainedunchanged and the y-coordinate has been multiplied by �

12�

The transformation is again a stretch. The scale factor ofthe stretch is �

12� and the direction of the stretch is parallel

to the y-axis.

y

2 4 6 8�2�4�6 xO

�2

�4

�6

2

4

6

8 y � x

x2y �

y � 2x

Solution 11a The graph of y � 2f(x) is drawn in red.

b The graph of y � �13�f(x) is drawn in blue.

The two diagrams show the graphs of y � f(x), y � f(�12�x) and y � f(2x) for the functions f(x) � x and

f(x) � x2

For any function, f, the transformation which maps the graph of y � f(x) onto the graph of y � f(ax),

for a � 0, is a stretch of scale factor �a1

� parallel to the x-axis.

y

2 4 6�2�4�6 xO

5

10

y � x 2

x2

4y �

y � 4 x2

These diagrams show stretches which are parallelto the x-axis.

Points on the graph of y � x are mapped ontocorresponding points on the graph of y � �

12�x.

For example (1, 1) → (2, 1), (2, 2) → (4, 2) and(�1, �1) → (�2, �1).

Points on the graph of y � x2 are mapped ontocorresponding points on the graph of y � �

14�x2

For example, (1, 1) → (2, 1), (2, 4) → (4, 4) and(�1, 1) → (�2, 1).

In all cases the y-coordinate has remainedunchanged and the x-coordinate has beenmultiplied by 2This transformation is a stretch of scale factor 2parallel to the x-axis.

Points on the graph of y � x are mapped ontocorresponding points on the graph of y � 2xFor example (1, 1) → (�

12�, 1), (2, 2) → (1, 2) and

(�1, �1) → (��12�, �1).

Points on the graph of y � x2 are mapped ontocorresponding points on the graph of y � 4x2

For example (1, 1) → (�12�, 1), (2, 4) → (1, 4) and

(�1, 1) → (��12�, 1).

In all cases the y-coordinate has again remainedunchanged and the x-coordinate has beenmultiplied by �

12�

The transformation is again a stretch. The scale

factor of the stretch is �12� and the direction of the

stretch is parallel to the x-axis.

y

2 4�2 xO

4

�2

�4

�1

�3

2

3

1

y � 2x

12y � x

The transformation that maps y � f(x) onto y � 2f(x) is astretch of scale factor 2 parallel to the y-axis. So the x-coordinate of each point on the graph willremain the same but each y-coordinate will be multipliedby 2 In particular (0, 1) → (0, 2) (2, �3) → (2, �6) (4, 1) → (4, 2)

594


The transformation that maps y � f(x) onto y � �13�f(x) is a

stretch of scale factor �13� parallel to the y-axis.

So the x-coordinate of each point on the graph willremain the same but each y-coordinate will be multipliedby �

13� In particular

(0, 1) → (0, �13�) (2, �3) → (2, �1) (4, 1) → (4, �

13�)

595

36.5 Applying stretches CHAPTER 36

The graph drawn in black has the equation y � f(x).Describe the stretch that will map

a y � f(x) onto graph (a) b y � f(x) onto graph (b)

Solution 12a (6, 0) → (12, 0) (3, 9) → (6, 9) (5, 5) → (10, 5)

The y-coordinates remain the same each time while the x-coordinates are multiplied by 2The transformation is a stretch parallel to thex-axis of scale factor 2

b (6, 0) → (2, 0) (3, 9) → (1 , 9)The transformation is a stretch parallel to the x-axis of scale factor �

13�

Exercise 36E

O

4

6

y

2�2 4 6 8 x

2

�2

O

4

y

2�2 4 6 x

2

�2

�4

�6

2 On the resource sheet draw on the sameaxes of the graph below the graphs of

a y � f(�12�x) b y � f(4x)

1 On the resource sheet draw on the sameaxes of the graph below the graphs of

a y � 3f(x) b y � �14�f(x)

A stretch of scale factor �a1

� parallel to the

x-axis maps the graph of y � f(x) onto thegraph of y � f(ax)

(b)

(a)

O

10

y

5 10 x

5

Example 12

596


3 The graph of y � f(x) is shown in black.

Describe the stretch that will map

a y � f(x) onto the blue graph labelled (a) b y � f(x) onto the red graph labelled (b)

4 The graph of y � x(x � 2) is stretched by a scale factor of 3 parallel to the y-axis.Write down the equation of the new graph.

5 The graph of y � x2 is stretched by a scale factor of 3 parallel to the x-axis.Write down the equation of the new graph.

6 The graph of y � f(x) where f(x) � x2 is stretched. Write down the equation of the new graphwhen the stretch is

a parallel to the y-axis with a scale factor of �12�

b parallel to the x-axis with a scale factor of �12�

7 The graph of y � f(x) has a vertex at (5, �3).Write down the coordinates of the vertex of

a y � 2f(x) b y � �12�f(x)

c y � f(3x) d y � f(�13�x)

36.6 Transformations applied to the graphs of sin x and cos xThe graphs of y � sin x° and y � cos x° for –360 � x � 360 were introduced in Section 31.2 and areshown below.

All the transformations described in the previous sections in this chapter can also be applied to thegraphs of y � sin xo and y � cos xo.

�0.5

�1

O

0.5

1

y

y � sin x°

�360 �270 �180 �90 90 180 270 360 x

y � cos x°

�1

�2

�3

�4

�5

�6

�7

O

1

y

�2 5

(a) (b)

x�1

�2

�3

�4

�5

�6

�7

O

1

y

�2 5 x

597

36.6 Transformations applied to the graphs of sin x and cos x CHAPTER 36

A sketch of the graph y � sin xo for 0 � x � 360 is shown.Sketch the graphs of

a y � sin x° � 2 b y � sin 2xo

Solution 13a

b

A sketch of the curve y � cos x° for 0 � x � 360 is drawn in black.

Using the sketch or otherwisefind the equations of thegraphs labelled (a) and (b).

Example 14

�1

O

1y

y � sin 2x°

90 180 270 360 x

�1

O

1

2

3

y

y � sin x° � 2

90 180 270 360 x

Example 13

�2

�1

O

1

y

(b)

(a)

30 60 90 120 150 180 210 240 270 300 330 360 x

�0.5

�1

O

0.5

1

y

y � sin x°

180 360 x

The transformation that will map the graph ofy � sin xo onto the graph of y � sin x° � 2 is atranslation of 2 units vertically in the positivey direction.

So the y-coordinate of every point on the graphof y � sin x° will increase by 2, in particular

(0, 0) → (0, 2) (90, 1) → (90, 3)(180, 0) → (180, 2) (270, �1) → (270, 1)(360, 0) → (360, 2)

The transformation that will map the graph of y � sin xo onto the graph of y � sin 2x°is a stretch, scale factor �

12� parallel to the x-axis.

So, the x-coordinate of every point on thegraph y � sin x° will be halved, in particular

(0, 0) → (0, 0) (90, 1) → (45, 1)(180, 0) → (90, 0)(270, �1) → (135, �1)(360, 0) → (180, 0)

Solution 14a y � cos(x � 30)o

b y � cos xo � 1

Describe fully the sequence of transformations that maps the graph of y � sin x° onto the graph of y � �

12� sin(x � 60)°.

Solution 15

Translation of � �(or 60 units horizontally in the negative x direction).

A stretch of scale factor �12� parallel

to the y-axis.

Exercise 36F

1 a Sketch the graph y � sin x° for 0 � x � 360

b On the same set of axes sketch the graphs ofi y � 2sin xo ii y � �sin xo

2 a Sketch the graph y � cos x° for 0 � x � 360

b On the same set of axes sketch the graphs ofi y � cos �

12�x° ii y � 2 � cos x°

3 Describe fully a sequence of transformations that maps the graph of y � sin x onto the graph of

a y � 5 sin(x � 30) b y � �sin 2x c y � �13�sin (–x)

4 A sketch of the curve y � cos x° for 0 � x � 360 is drawn in black.Using the sketch or otherwise find theequations of the graphs labelled i and ii.

5 The maximum possible value of sin x° is 1 Write down the maximum possible value of

a 4 sin x° b sin x° � 2 c sin x° � 1 d �14� sin x°

e sin(x � 30)o f 5 sin x° � 2 g sin 3x° h �12� sin x° � 3

�2

�3

�1

O

1

2

3y

(ii)

(i)

90 180 270 360 x

Next consider the mapping of y � sin(x � 60)o onto y � �

12�sin(x � 60)°.

This is a transformation of the type that maps y � f(x) onto y � af(x) so is a stretch of scale factor a parallel to the y-axis.

First consider the mapping of y � sin xo onto y � sin(x � 60)o

This is a transformation of the type that maps y � f(x) onto y � f(x � a) so is a translation of a units horizontally in thenegative x direction.

�600

Example 15

The transformation that would map the graph of y � cos xo onto the bluegraph is a translation of 1 unit vertically in the negative y direction.

598


The transformation that would map the graph of y � cos xo onto the red graph

is a translation of � �(or 30 units horizontally in the positive x direction).300

599

Chapter 36 review questions CHAPTER 36

Chapter summary

Chapter 36 review questions1 Given that f(x) � (x � 4)2 work out the value of

a f(0) b f(5) c f(�6)

2 Given that g(x) � 5 � x write down an expression for

a g(2x) b g(�x) c g(x � 2) d g(a � x)

3 This is a sketch of the curve with equation y � f(x).

The vertex of the curve is A(2, 12).Write down the coordinates of the vertex for each of the curves having the following equations.

a y � f(x) � 6 b y � f(x � 3) c y � f(�x) d y � f(4x)(1385 June 1999)

O

y A(2, 12)

x

You should now know that:

a function is a rule that can be used to change numbers

f(x) is read as ‘f of x’ and means that f is a function of x

f(3) means work out the value of f(x) when x � 3

function notation can be used when drawing graphs

for any function, f, the transformation which maps the graph of

y � f(x) onto the graph of y � f(x) � a is a translation of � �.If a � 0 this is a translation in the positive y direction If a � 0 this is a translation in the negative y direction

for any function, f, the transformation which maps the graph of

y � f(x) onto the graph of y � f(x � a) is a translation of � �.If a � 0 this is a translation in the negative x direction.If a � 0 this is a translation in the positive x direction.

for any function, f, the transformation which maps the graph ofy � f(x) onto the graph of y � �f(x) is a reflection in the x-axisy � f(x) onto the graph of y � f(�x) is a reflection in the y-axis

for any function, f, the transformation which maps the graph ofy � f(x) onto the graph of y � af(x) is a stretch of scale factor a parallel to the y-axis

y � f(x) onto the graph of y � f(ax) is a stretch of scale factor �a1

� parallel to the x-axis

All the above transformations can be applied to the graphs of y � sin x° and y � cos x°

�a0

0a

�

�

�

�

�

�

�

�

�

�

4 This is a sketch of the curve with equation y � f(x).It passes through the origin O.The only vertex of the curve is at A (2, �4).

a Write down the coordinates of the vertex of the curve with equationi y � f(x � 3) ii y � f(x) � 5 iii y � �f(x) iv y � f(2x)

The curve with equation y � x2 has been translated to give the curve y � f(x).

b Find f(x) in terms of x. (1387 June 2003)

5 A transformation has been applied to the graph of y � x2 to give the graph of y � �x2

a Describe fully the transformation.

For all values of x

x2 � 4x � (x � p)2 � q

b Find the values of p and q.

A transformation has been applied to the graph of y � x2 to give the graph of y � x2 � 4x.

c Using your answer to part b, or otherwise, describe fully the transformation.(1385 June 2001)

6 The expression x2 � 6x � 14 can be written in the form (x � p)2 � q for all values of x.

a Find the value of i p ii qThe equation of a curve is y � f(x) where f(x) � x2 � 6x � 14Here is a sketch of the graph of y � f(x).

b Write down the coordinates of the minimum point, M, of the curve.

Here is a sketch of the graph of y � f(x) � k where k is a positive constant. The graph touchesthe x-axis.

c Find the value of k. d For the graph of y � f(x � 1)i write down the coordinates of the minimum point

ii find the coordinates of the point where the curve crosses the y-axis. (1387 November 2003)

O

y

y � f(x) � k

x

O

y

M

y � f(x)

x

O

y

A(2, �4)

x

y � f(x)

600


601

Chapter 36 review questions CHAPTER 36

7 A sketch of the curve y � sin x° for 0 � x � 360 is shown on the right.

a Using the sketch above, or otherwise, find the equation of each of the following two curves.i ii

b Describe fully the sequence of two transformations that maps the graph of y � sin x° ontothe graph of y � 3 sin 2x°. (1387 June 2004)

8 The graph of y � a � b cos (kt) for values of t between 0°and 120°, is drawn on the grid.

Use the graph to find an estimate for the value ofi a

ii biii k.

(1387 June 2003)

9 a On the grid on the resource sheet sketch the graphs of i y � sin x°

ii y � sin 2x°for values of x between 0 and 360 Label each graph clearly.

b Calculate all the solutions to the equation

2 sin 2x° � �1

between x � 0 and x � 360

(1385 June 1998)�2

�1

0

1

2

360

y

x

20

40

60

80

100

O 30° 60° 90° 120°

y

y � a � b cos(kt)

t

O

y

x

�1

�2

1

2

90 180 270 360O

y

x

�1

�2

1

2

90 180 270 360

O

y

x

�1

�2

1

2

90 180 270 360

CHAPTER 36... · 584 CHAPTER 36 Transformations of functions So the transformation that maps the...

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Transcript of CHAPTER 36... · 584 CHAPTER 36 Transformations of functions So the transformation that maps the...