Transformation Rules

TransformationsTransformations

Transformation Rules

Translations applied to a graph mean move movements

Horizontally or Vertically or both

5

2

They can be described using a vector as shown below

Means translate(move) a graph 5 to the right And translate (move it) 2 up

-1

1

2

3

4

5

-1-2 1 2 3 4 5 6 70X->

|̂Y

Transformation Rules

2xy

The graph of forms a curve called a parabola

2xy

This point . . . is called the vertex

Transformation Rules

32 xy2xy

2xy

Adding a constant translates up the y-axis

2xy 32 xye.g.

2xy

The vertex is now ( 0, 3)

has added 3 to the y-values

2xy 32 xy

Transformation Rules

We get

Adding 3 to x gives 23)( xy2xy

Adding 3 to x moves the curve 3 to the left.

23 )( xy

2xy

Transformation Rules Translating in both

directions 35 2 )(xy2xy e.g.

3

5

We can write this in vector form as:

translation

35 2 )(xy2xy

Transformation Rules

SUMMARY

The curve

is a translation of by 2xy

q

pqpxy 2)(

The vertex is given by ),( qp

Transformation Rules

SUMMARY

Memory Aid

HIVO – Horizontal Inside the Bracket

Vertical Outside the bracket

HOVIS – Horizontal Opposite of what it says

Vertical Is Same as what it says

Transformation RulesExercises: Sketch the following translations of 2xy

12 2 )(xy2xy 1.

23 2 )(xy2xy 2.

34 2 )(xy2xy 3.

1)2( 2 xy

2xy

2xy

2)3( 2 xy

2xy

3)4( 2 xy

Transformation Rules

4 Sketch the curve found by translating2xy

3

2

2

12xy

by . What is its equation?

5 Sketch the curve found by translating

by . What is its equation?

32 2 )(xy

21 2 )(xy

Transformation Rules

3xy 1)2( 3 xy

e.g. The translation of the function by

the vector gives the function

.

3xy

1

21)2( 3 xy

The graph becomes

1

2

Means translate horizontally by +2.So put –2 in the bracket.

HIvo

HOvis

Transformation Rules

3xy 1)2( 3 xy

e.g. The translation of the function by

the vector gives the function

.

3xy

1

21)2( 3 xy

The graph becomes

1

2

Means translate vertically by +1.So put +1 outside the bracket.

HIVOHOVIS

Transformation Rules

e.g.1 Consider the following functions:2xy an

d

24xy

For yxxy 2,2 4

yxxy 2,4 2For 16

In transforming from to the y-value has been multiplied by 4

2xy 24 xy

Vertical Stretches

Transformation Rules

e.g.1 Consider the following functions:2xy an

d

24xy

For yxxy 2,2

yxxy 2,4 2For

Similarly, for every value of x, the y-value on

is 4 times the y-value on 24xy 2xy

22 4 xyxy is a stretch of scale factor 4 parallel to the y-axis

In transforming from to the y-value has been multiplied by 4

2xy 24 xy

4

16

Transformation Rules

e.g.1 Consider the following functions:2xy an

d

24xy

HIVO – Horizontal Inside the Bracket

Vertical Outside the bracket

HOVIS – Horizontal Opposite of what it says

Vertical Is Same as what it says

As the 4 is Outside the stretch is applied

Vertically

Transformation Rules

2xy

24 xy

The graphs of the functions are as follows:

)4,1(

)1,1(

2xy is a stretch of

24xy by scale factor 4, parallel to the y-

axis

Transformation Rules

2xy

24 xy

is a transformation of given by 24xy 2xy

a stretch of scale factor 4 parallel to the y-axis

4

Transformation Rules

-1

1

2

3

4

5

-1-2 1 2 3 4 5 6 70X->

|̂Y

Now, for

2y x , x y 9 3

2y (3x) , x y 9 and for

1

The x-value must be divided by 3 to give the same value of y.

Horizontal stretches

So the x coordinate is stretched by a factor of parallel to the x axis

2y x

2y (3x)

Transformation Rules

e.g.1 Consider the following functions:2xy an

d

2y (3x)

HIVO – Horizontal Inside the Bracket

Vertical Outside the bracket

HOVIS – Horizontal Opposite of what it says

Vertical Is Same as what it says

As the 3 is Inside the stretch is applied

Horizontally and is the Opposite. So instead of being 3 times as wide it is a as wide.

Transformation Rules

22 4 xyxy is a stretch of scale factor 4 parallel to the y-axis

or

22 )2( xyxy is a stretch of scale factor parallel to the x-axis21

2xy 24 xy The transformation of to

SUMMARY

Transformation Rules

SUMMARY 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

Transformation Rules

We always stretch from an axis.

xy

1

xy

3

Using the same axes, sketch both functions.

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

3

Transformation Rules

(b) 2xy

29 xy

Exercises1. (a) Describe a transformation of

that gives .

2xy 29xy

(b) Sketch the graphs of both functions to illustrate your answer.

Solutio

n

:

(a) A stretch of s.f. 9 parallel to the y-axis.

Transformation Rules

Transforming the exponential graph y = ex

-1-2-3-4-5

12345

-1-2-3-4 1 2 3 4 50

Transformation Rules

-1

-2

-3

-4

-5

1

2

3

4

5

-1-2-3-4 1 2 3 4 50

y = ex y = e(2x)

Horizontal stretch scale factor

All the x coordinates are as wide

-1

-2

-3

-4

-5

1

2

3

4

5

-1-2-3-4 1 2 3 4 50

Transformation Rules

y = ex

-1

-2

-3

-4

-5

1

2

34

5

-1-2-3-4 1 2 3 4 50

y = 2e(2x)

Horizontal stretch scale factor Vertical stretch scale factor 2 All the y coordinates are 2x as high

-1

-2

-3

-4

-5

1

2

3

4

5

-1-2-3-4 1 2 3 4 50

Transformation Rules

-1

-2

-3

-4

-5

1

2

34

5

-1-2-3-4 1 2 3 4 50

Horizontal stretch scale factor Vertical stretch scale factor 2

Vertical translation0

4

y = 2e(2x)–4

-1

-2

-3

-4

-5

1

2

34

5

-1-2-3-4 1 2 3 4 50

y = ex

Transformation Rules

Two more Transformations Reflection in the x-

axis Every y-value changes sign when we reflect in the x-axis e.g.

So, y x y x2 2( )

In general, a reflection in the x-axis is given by

)()( xfyxfy

-1

-2

-3

-4

-5

1

2

3

4

5

-1-2-3-4 1 2 3 4 50X->

|̂Y

-1

-2

-3

-4

-5

1

2

3

4

5

-1-2-3-4 1 2 3 4 50X->

|̂Y

y=x2

y=–(x)2

Transformation Rules Reflection in the y-

axis Every x-value changes sign when we reflect in the y-axis e.g.

y x y x3 3( )

In general, a reflection in the y-axis is given by

)()( xfyxfy

-1

-2

-3

-4

-5

1

2

3

4

5

-1-2-3-4 1 2 3 4 50X->

|̂Y

y x 3( )

-1

-2

-3

-4

-5

1

2

3

4

5

-1-2-3-4 1 2 3 4 50X->

|̂Y

y x 3

Transformation Rules

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