combined transformation

7/29/2019 Combined Transformation

1/18

Form 5 Chapter 3: Transformations III


2/18

Determining the image of an object undercombination of two isometric transformations

Diagram P Diagram Q DiagramR

TransformationBA

TransformationA TransformationB


3/18

Example

TransformationA = translation

TransformationB = translation

Determine the image of the triangle P under the combined transformationBA.

x

y

2 4 6

2

4

6

6

4

2 2 4 6

P

O


4/18

Example

Transformation A = translation

Transformation B= translation

The image of the triangle P under the combined transformationBA is

the triangleI.

Solution:

x

y

2 4 6

2

4

6

6

4

2 2 4 6

P

P

P

I

O


5/18

Determining the image of an objectunder combination of two enlargements

E= enlargement at V(1, 7) with a

scale factor of 2F= enlargement at W(3, 7) with a

scale factor of 2

Determine the image of the

rectangle P under the combined

transformation FE.

x

y

2 4 6

2

4

6

6

4

2 2 4 6

P

O


6/18

Determining the image of an objectunder combination of two enlargements


scale factor of 2

The image of the rectangle P under

the combined transformation FEisthe rectangleI.

Solution:

x

y

2 4 6

2

4

6

6

4

2 2 4 6

P

V

P

W

I

O

F= enlargement at W(3, 7) with a

scale factor of2

1


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Determine the image of an object under combinationof an enlargement and an isometric transformation


scale factor of 2H= reflection in the linex = 0

Determine the image of the

rectangle P under the combined

transformationEH.

x

y

2 4 6

2

4

6

6

4

2 2 4 6

P

O


8/18

Determine the image of an object under combinationof an enlargement and an isometric transformation


scale factor of 2H= reflection in the linex = 0

The image of the rectangle P under

the combined transformationEHisthe rectangleII.

Solution:

x

y

2 4 6

2

4

6

6

4

2 2 4 6

PP

II

V

O

P


9/18

Drawing the image for combinationof two transformations

A = reflection in the liney = 1

B = rotation through 90 clockwise

about (1, 2)

Draw the image of the triangle PQR

under the combined transformation

BA.

x

y

2 4 6

2

4

6

6

4

2 2 4 6

P

Q

R

O


10/18


B = rotation through 90 clockwise

about (1, 2)

Solution:

x

y

2 4 6

2

4

6

6

4

2 2 4 6

P

Q

R

y = 1

R

P

QP

Q

R

O

Drawing the image for combinationof two transformations


11/18

Stating the coordinates of the imageunder combined transformation

Step 1: Determine the coordinates ofM'

image ofM, under the first transformationB.

Step 2: Determine the coordinates ofM''

image ofM', under the second transformationA.


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Summary I:

PointM(x,y)Image:

M'= (x +h,y +k)Translation =

PointM(x,y)Image:

M'= (x, y)Reflection =x-axis

PointM(x,y)Image:

M'= (x,y)

Reflection =y-axis


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Summary II:

PointM(x,y)Image:

M'= (y, x)

Rotation:

rotation through

90 clockwise at

the origin

PointM(x,y)Image:

M'= (

y,x)

Rotation:

rotation through

90

anticlockwiseat the origin


14/18

Summary III:

PointM(x,y)Image:

M'= (x,y)

Rotation:

rotation 180 at

the origin

PointM(x,y)Image:

M'= (kx,ky)

Enlargement:

enlargement at the

origin with a scalefactor ofk


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B = rotation through 90 anticlockwise

at the origin

State the coordinates of the image

of pointMunder the combined

transformationAB.

Stating the coordinates of the image undercombined transformationAB

x

y

2 4 6

2

4

6

6

4

2 2 4 6

PM(6, 4)


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Step 1:

Transformation B:

Rotation through 90

anticlockwise at the origin

Solution:

PointM(6, 4)

Image:

M'= (4, 6)

x

y

2 4 6

2

4

6

6

4

2 2 4 6

PM(6, 4)

M' (4, 6)


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Step 2:

TransformationA:

Reflection in the line

y = 1

Solution:

PointM'(4, 6)

Image:

M''= (4, 4)

The image of Munder combinedtransformation AB= M'' = (4, 4)


x

y

2 4 6

2

4

6

6

4

2 2 4 6

P

M'' (4, 4)

M' (

4, 6)

y = 1

M(6, 4)


18/18

Specifying the equivalent of a combination oftwo isometric transformationsII= a rotation of 180 about the centre (2, 3)

P P' P''

Solution:

x

y

2 4 6

2

4

6

6

4

2 2 4 6

P

P''

P'

I

IISingle

transformationZ

Single transformationZ= a rotation of 180 about

the origin

I= Translation

combined transformation

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