combined transformation
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Form 5 Chapter 3: Transformations III
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Determining the image of an object undercombination of two isometric transformations
Diagram P Diagram Q DiagramR
TransformationBA
TransformationA TransformationB
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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
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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
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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
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Determining the image of an objectunder combination of two enlargements
E= enlargement at V(1, 7) with a
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
E= enlargement at V(1, 7) with a
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
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Determine the image of an object under combinationof an enlargement and an isometric transformation
E= enlargement at V(1, 7) with a
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
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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
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A = reflection in the liney = 1
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
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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
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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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A = reflection in the liney = 1
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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Stating the coordinates of the image undercombined transformationAB
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)
Stating the coordinates of the image undercombined transformationAB
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)
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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
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