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Tricks in Transformations c- [email protected]
Dr D K R Babajee
Mauritius
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Transformations
We require compass, protractor and ruler to find thetransformations of points.
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Transformations
We require compass, protractor and ruler to find thetransformations of points.
In this presentation, we show how to get the reflection,
rotation, enlargement, stretch and shear using sometricks without the use of ruler, protractor and compass.
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Part 1: Reflection
Reflection in the x-axis
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Part 1: Reflection
Reflection in the x-axis Reflection in the horizontal line
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Part 1: Reflection
Reflection in the x-axis Reflection in the horizontal line
Reflection in the y-axis
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Part 1: Reflection
Reflection in the x-axis Reflection in the horizontal line
Reflection in the y-axis Reflection in the vertical line
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Part 1: Reflection
Reflection in the x-axis Reflection in the horizontal line
Reflection in the y-axis Reflection in the vertical line
Reflection in the line y=x
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Part 1: Reflection
Reflection in the x-axis Reflection in the horizontal line
Reflection in the y-axis Reflection in the vertical line
Reflection in the line y=x Reflection in the line
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fl
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Part 1: Reflection
Reflection in the x-axis Reflection in the horizontal line
Reflection in the y-axis Reflection in the vertical line
Reflection in the line y=x Reflection in the line
Reflection in the line y=-x
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P 1 R fl i
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Part 1: Reflection
Reflection in the x-axis Reflection in the horizontal line
Reflection in the y-axis Reflection in the vertical line
Reflection in the line y=x Reflection in the line
Reflection in the line y=-x Reflection in the line
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P t 1 R fl ti
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Part 1: Reflection
Reflection in the x-axis
Reflection in the horizontal line
Reflection in the y-axis
Reflection in the vertical line
Reflection in the line y=x Reflection in the line
Reflection in the line y=-x Reflection in the line
Finding the mirror line (horizontal or vertical line)
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R fl ti i th i ( 0)
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Reflection in the x-axis (y=0)
The matrix of this transformation is given by
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Reflection in the a is ( 0)
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Reflection in the x-axis (y=0)
The matrix of this transformation is given by
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Reflection in the x axis (y 0)
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Reflection in the x-axis (y=0)
The matrix of this transformation is given by
Reflection
in x-axis
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Reflection in the x axis (y 0)
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Reflection in the x-axis (y=0)
The matrix of this transformation is given by
Reflection
in x-axis
-coordinate is unchanged and we reverse the sign ofy-coordinate.
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Example 1 1
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Example 1.1
Find the reflection of the triangle having the coordinates ,
and C in the x-axis.
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Example 1 1
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Example 1.1
Find the reflection of the triangle having the coordinates ,
and C in the x-axis.
reverse sign of coor.
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Reflection in the horizontal line
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Reflection in the horizontal line
We must change the point relative to the x-axis .
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Reflection in the horizontal line
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Reflection in the horizontal line
We must change the point relative to the x-axis .
We subtract from the y-coordinate to get Change
to line .
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Reflection in the horizontal line
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Reflection in the horizontal line
We must change the point relative to the x-axis .
We subtract from the y-coordinate to get Change
to line .
Then we apply the rule to reflection in x-axis.
Reflection
in x-axis
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Reflection in the horizontal line
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Reflection in the horizontal line
We must change the point relative to the x-axis .
We subtract from the y-coordinate to get Change
to line .
Then we apply the rule to reflection in x-axis.
Reflection
in x-axis
Then we add to the resulting y-coordinate to return to the line
Change
to line
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Reflection in the horizontal line
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Reflection in the horizontal line
We must change the point relative to the x-axis .
We subtract from the y-coordinate to get Change
to line .
Then we apply the rule to reflection in x-axis.
Reflection
in x-axis
Then we add to the resulting y-coordinate to return to the line
Change
to line
subtract
from
reverse sign
of
add
to
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Example 1.2
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Example 1.2
Find the reflection of the triangle having the coordinates
,
and C in the line .
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Example 1.2
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p
Find the reflection of the triangle having the coordinates
,
and C in the line .
subtract
from
coor. reverse sign of
coor. add 1 to
coor.
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Reflection in the y-axis (x=0)
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y ( )
The matrix of this transformation is given by
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Reflection in the y-axis (x=0)
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y ( )
The matrix of this transformation is given by
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Reflection in the y-axis (x=0)
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y ( )
The matrix of this transformation is given by
Reflection
in y-axis
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Reflection in the y-axis (x=0)
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The matrix of this transformation is given by
Reflection
in y-axis
we reverse the sign of x-coordinate and -coordinate isunchanged.
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Example 1.3
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Find the reflection of the triangle having the coordinates ,
and C in the y-axis.
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Example 1.3
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Find the reflection of the triangle having the coordinates ,
and C in the y-axis.
reverse sign of coor.
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Reflection in the vertical line
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We must change the point relative to the y-axis, .
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Reflection in the vertical line
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We must change the point relative to the y-axis, .
We subtract from the x-coordinate to get Change
to line .
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Reflection in the vertical line
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We must change the point relative to the y-axis, .
We subtract from the x-coordinate to get Change
to line .
Then we apply the rule to reflection in y-axis.
Reflection
in y-axis
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Reflection in the vertical line
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We must change the point relative to the y-axis, .
We subtract from the x-coordinate to get Change
to line .
Then we apply the rule to reflection in y-axis.
Reflection
in y-axis
Then we add to the resulting x-coordinate to return to the line
Change
to line
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Reflection in the vertical line
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We must change the point relative to the y-axis, .
We subtract from the x-coordinate to get Change
to line .
Then we apply the rule to reflection in y-axis.
Reflection
in y-axis
Then we add to the resulting x-coordinate to return to the line
Change
to line
subtract
from
reverse sign
of
add
to
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Example 1.4
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Find the reflection of the triangle having the coordinates , and
C in the line .
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Example 1.4
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Find the reflection of the triangle having the coordinates , and
C in the line .
subtract
from
coor. (+1) reverse sign of
coor. add -1 to
coor. (-1)
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Figure for Reflection for Examples 1.1 to 1.4
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Reflection in the line y=x
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The matrix of this transformation is given by
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Reflection in the line y=x
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The matrix of this transformation is given by
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Reflection in the line y=x
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The matrix of this transformation is given by
Reflection
in y=x
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Reflection in the line y=x
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The matrix of this transformation is given by
Reflection
in y=x
Interchange the coordinates of and .
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Example 1.5
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Find the reflection of the triangle having the coordinates ,
and C in the line .
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Example 1.5
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Find the reflection of the triangle having the coordinates ,
and C in the line .
Interchange
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Reflection in the line
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We must change the point relative to the line .
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Reflection in the line
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We must change the point relative to the line .
We subtract from the y-coordinate to get Change
to line .
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Reflection in the line
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We must change the point relative to the line .
We subtract from the y-coordinate to get Change
to line .
Then we apply the rule to reflection in the line .
Reflection
in line y=x
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Reflection in the line
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We must change the point relative to the line .
We subtract from the y-coordinate to get Change
to line .
Then we apply the rule to reflection in the line .
Reflection
in line y=x
Then we add to the resulting y-coordinate to return to the line
Change
to line
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Reflection in the line
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We must change the point relative to the line .
We subtract from the y-coordinate to get Change
to line .
Then we apply the rule to reflection in the line .
Reflection
in line y=x
Then we add to the resulting y-coordinate to return to the line
Change
to line
subtract
from
interchange
x and y-c
add
to new y coor.
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Example 1.6
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Find the reflection of the triangle having the coordinates , and
C in the line .
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Example 1.6
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Find the reflection of the triangle having the coordinates , and
C in the line .
subtract
from coor. interchange add 1 to coor.
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Reflection in the line y=-x
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The matrix of this transformation is given by
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Reflection in the line y=-x
Th i f hi f i i i b
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The matrix of this transformation is given by
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Reflection in the line y=-x
Th t i f thi t f ti i i b
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The matrix of this transformation is given by
Reflection
in y=-x
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Reflection in the line y=-x
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The matrix of this transformation is given by
Reflection
in y=-x
Interchange the coordinates of
and
and reversetheir signs.
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Example 1.7
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Find the reflection of the triangle having the coordinates ,
and C in the line .
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Example 1.7
Fi d h fl i f h i l h i h di
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Find the reflection of the triangle having the coordinates ,
and C in the line .
Interchange Reverse signs of both coor.
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Reflection in the line
W t h th i t l ti t th li
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We must change the point relative to the line .
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Reflection in the line
We must change the point relative to the line
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We must change the point relative to the line .
We subtract from the y-coordinate to get Change
to line .
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Reflection in the line
We must change the point relative to the line
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We must change the point relative to the line .
We subtract from the y-coordinate to get Change
to line .
Then we apply the rule to reflection in the line .
Reflection
in line y=-x
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Reflection in the line
We must change the point relative to the line
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We must change the point relative to the line .
We subtract from the y-coordinate to get Change
to line .
Then we apply the rule to reflection in the line .
Reflection
in line y=-x
Then we add to the resulting y-coordinate to return to the line
Change
to line
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Reflection in the line
We must change the point relative to the line
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We must change the point relative to the line .
We subtract from the y-coordinate to get Change
to line .
Then we apply the rule to reflection in the line .
Reflection
in line y=-x
Then we add to the resulting y-coordinate to return to the line
Change
to line
subtract
from
interchange and reverse signs
add
to new y coor.
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Example 1.8
Find the reflection of the triangle having the coordinates and
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Find the reflection of the triangle having the coordinates , and
C in the line .
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Example 1.8
Find the reflection of the triangle having the coordinates and
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Find the reflection of the triangle having the coordinates , and
C in the line .
subtract
from
coor. (+1) interchange and reverse signs add
to
coor. (-1)
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Figure for Reflection for Examples 1.5 to 1.8
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Finding the mirror line (horizontal or vertical line)
The line of reflection (mirror line) is the perpendicular bisector of the object and
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e e o e ect o ( o e) s t e pe pe d cu a b secto o t e object a d
the image.
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Finding the mirror line (horizontal or vertical line)
The line of reflection (mirror line) is the perpendicular bisector of the object and
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( ) p p j
the image.
We show a trick to find the mirror line in case of a vertical or horizontal.
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Example 1.9
Describe the transformation when the triangle
is mapped on triangle
:
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g pp g
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Example 1.9
Describe the transformation when the triangle
is mapped on triangle
:
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Example 1.9
Describe the transformation when the triangle
is mapped on triangle
:
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We observe that y-coor. are unchanged and so it is a reflection in the line .
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Example 1.9
Describe the transformation when the triangle
is mapped on triangle
:
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We observe that y-coor. are unchanged and so it is a reflection in the line .
To find , we find the x-coor. of the midpoint of the object and image.
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Example 1.9
Describe the transformation when the triangle
is mapped on triangle
:
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We observe that y-coor. are unchanged and so it is a reflection in the line .
To find , we find the x-coor. of the midpoint of the object and image.
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Example 1.9
Describe the transformation when the triangle
is mapped on triangle
:
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We observe that y-coor. are unchanged and so it is a reflection in the line .
To find , we find the x-coor. of the midpoint of the object and image.
Reflection in the line
.
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Example 1.10
Describe the transformation when the triangle
is mapped on triangle
:
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Example 1.10
Describe the transformation when the triangle
is mapped on triangle
:
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Example 1.10
Describe the transformation when the triangle
is mapped on triangle
:
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We observe that x-coor. are unchanged and so it is a reflection in the line .
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Example 1.10
Describe the transformation when the triangle
is mapped on triangle
:
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We observe that x-coor. are unchanged and so it is a reflection in the line .
To find , we find the y-coor. of the midpoint of the object and image.
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Example 1.10
Describe the transformation when the triangle
is mapped on triangle
:
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We observe that x-coor. are unchanged and so it is a reflection in the line .
To find , we find the y-coor. of the midpoint of the object and image.
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Example 1.10
Describe the transformation when the triangle
is mapped on triangle
:
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We observe that x-coor. are unchanged and so it is a reflection in the line .
To find , we find the y-coor. of the midpoint of the object and image.
Reflection in the line
.
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Figure for Reflection for Examples 1.9 and 1.10
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Part 2: Rotation
Rotation about the origin
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Part 2: Rotation
Rotation about the origin
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Rotation about the point
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Part 2: Rotation
Rotation about the origin
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Rotation about the point
Anticlockwise Rotation about the origin
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Part 2: Rotation
Rotation about the origin
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Rotation about the point
Anticlockwise Rotation about the origin
Anticlockwise Rotation about the point
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Part 2: Rotation
Rotation about the origin
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Rotation about the point
Anticlockwise Rotation about the origin
Anticlockwise Rotation about the point
Clockwise Rotation about the origin
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Part 2: Rotation
Rotation about the origin
R i b h i
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Rotation about the point
Anticlockwise Rotation about the origin
Anticlockwise Rotation about the point
Clockwise Rotation about the origin
Clockwise Rotation about the point
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Rotation about the origin
The matrix of this transformation is given by
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Rotation about the origin
The matrix of this transformation is given by
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Rotation about the origin
The matrix of this transformation is given by
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Rotation
about origin
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Rotation about the origin
The matrix of this transformation is given by
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Rotation
about origin
We reverse the signs of both x- and y-coordinates.
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Example 2.1
Find the rotation of the triangle having the coordinates
and C about the origin
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, and C about the origin.
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Example 2.1
Find the rotation of the triangle having the coordinates
and C about the origin
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, and C about the origin.
reverse signs of
and
coor.
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Rotation about the point
We must change the point relative to the origin so that the centre is the
origin
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origin.
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Rotation about the point
We must change the point relative to the origin so that the centre is the
origin
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origin.
We subtract from to get Change
to origin .
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Rotation about the point
We must change the point relative to the origin so that the centre is the
origin
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origin.
We subtract from to get Change
to origin .
Then we apply the rule to
rotation about origin.
Rotation
about origin
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Rotation about the point
We must change the point relative to the origin so that the centre is the
origin.
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origin.
We subtract from to get Change
to origin .
Then we apply the rule to
rotation about origin.
Rotation
about origin
Then we add to the resulting coordinates to return to the point
Change
to point
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Rotation about the point
We must change the point relative to the origin so that the centre is the
origin.
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origin.
We subtract from to get Change
to origin .
Then we apply the rule to
rotation about origin.
Rotation
about origin
Then we add to the resulting coordinates to return to the point
Change
to point
subtract
reverse signs of
x and y coor.
add
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Example 2.2
Find the rotation of the triangle having the coordinates ,
and C
about the point
.
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and C about the point .
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Example 2.2
Find the rotation of the triangle having the coordinates ,
and C
about the point
.
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a d C about t e po t
subtract
reverse signs of
and
coor. add
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Anticlockwise Rotation about the origin
The matrix of this transformation is given by
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Anticlockwise Rotation about the origin
The matrix of this transformation is given by
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The matrix of this transformation is given by
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Anticlockwise Rotation about the origin
The matrix of this transformation is given by
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The matrix of this transformation is given by
Anticlockwise
Rotation about origin
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Anticlockwise Rotation about the origin
The matrix of this transformation is given by
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e at o t s t a s o at o s g e by
Anticlockwise
Rotation about origin
We reverse the sign of y-coordinate and interchangethe resulting coordinates.
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Example 2.3
Find the anticlockwise rotation of the triangle having the
coordinates , and C about the origin.
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g
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Example 2.3
Find the anticlockwise rotation of the triangle having the
coordinates
,
and C
about the origin.
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reverse signs of coor. Interchange
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Anticlockwise about the point
We must change the point relative to the origin so that the centre is the
origin.
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Anticlockwise about the point
We must change the point relative to the origin so that the centre is the
origin.
Change
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We subtract from to get Change
to origin .
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Anticlockwise about the point
We must change the point relative to the origin so that the centre is the
origin.
W bt t f t tChange
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We subtract from to get Change
to origin .
Then we apply the rule to
anticlockwise rotation about origin.
anticlockwise
rotation about origin
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Anticlockwise about the point
We must change the point relative to the origin so that the centre is the
origin.
W bt t f t tChange
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We subtract from to get g
to origin .
Then we apply the rule to
anticlockwise rotation about origin.
anticlockwise
rotation about origin
Then we add
to the resulting coordinates to return to the point
Change
to point
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Anticlockwise about the point
We must change the point relative to the origin so that the centre is the
origin.
We subtract from to getChange
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We subtract from to get g
to origin .
Then we apply the rule to
anticlockwise rotation about origin.
anticlockwise
rotation about origin
Then we add
to the resulting coordinates to return to the point
Change
to point
subtract
reverse sign of y coor.
and interchange
add
(a,b)
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Example 2.4
Find the anticlockwise rotation of the triangle having the coordinates
,
and C
about the point
.
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Example 2.4
Find the anticlockwise rotation of the triangle having the coordinates
,
and C
about the point
.
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subtract
reverse sign of
coor. Interchange add
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Clockwise Rotation about the origin
The matrix of this transformation is given by
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Clockwise Rotation about the origin
The matrix of this transformation is given by
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Clockwise Rotation about the origin
The matrix of this transformation is given by
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Clockwise
Rotation about origin
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Clockwise Rotation about the origin
The matrix of this transformation is given by
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Clockwise
Rotation about origin
We reverse the sign of x-coordinate and interchangethe resulting coordinates.
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Example 2.5
Find the clockwise rotation of the triangle having the
coordinates
,
and C
about the origin.
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Tricks in Transformations c-
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Example 2.5
Find the clockwise rotation of the triangle having the
coordinates
,
and C
about the origin.
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reverse signs of coor. Interchange
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clockwise about the point
We must change the point relative to the origin so that the centre is the
origin.
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clockwise about the point
We must change the point relative to the origin so that the centre is the
origin.
We subtract from to get Change
t i i .
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gto origin
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clockwise about the point
We must change the point relative to the origin so that the centre is the
origin.
We subtract from to get Change
t i i .
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to origin
Then we apply the rule to
clockwise rotation about origin.
clockwise
rotation about origin
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clockwise about the point
We must change the point relative to the origin so that the centre is the
origin.
We subtract from to get Change
to origin .
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to origin
Then we apply the rule to
clockwise rotation about origin.
clockwise
rotation about origin
Then we add
to the resulting coordinates to return to the point
Change
to point
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clockwise about the point
We must change the point relative to the origin so that the centre is the
origin.
We subtract from to get Change
to origin .
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to origin
Then we apply the rule to
clockwise rotation about origin.
clockwise
rotation about origin
Then we add
to the resulting coordinates to return to the point
Change
to point
subtract
reverse sign of x coor.
and interchange
add
(a,b)
Tricks in Transformations c-
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Example 2.6
Find the clockwise rotation of the triangle having the coordinates
,
and C
about the point
.
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Example 2.6
Find the clockwise rotation of the triangle having the coordinates
,
and C
about the point
.
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subtract
reverse sign of
coor. Interchange add
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Figure for Rotation about the origin
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Figure for Rotation about the point
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Part 3: Enlargement, Stretch and Shear
Enlargement about origin with scale factor
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Part 3: Enlargement, Stretch and Shear
Enlargement about origin with scale factor
Enlargement about point
with scale factor
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Part 3: Enlargement, Stretch and Shear
Enlargement about origin with scale factor
Enlargement about point
with scale factor
Stretch parallel to -axis, invariant with stretch factor
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Part 3: Enlargement, Stretch and Shear
Enlargement about origin with scale factor
Enlargement about point
with scale factor
Stretch parallel to -axis, invariant with stretch factor
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Stretch parallel to -axis, invariant with stretch factor
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Part 3: Enlargement, Stretch and Shear
Enlargement about origin with scale factor
Enlargement about point
with scale factor
Stretch parallel to -axis, invariant with stretch factor
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Stretch parallel to -axis, invariant with stretch factor
Stretch parallel to -axis, invariant with stretch factor
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Part 3: Enlargement, Stretch and Shear
Enlargement about origin with scale factor
Enlargement about point
with scale factor
Stretch parallel to -axis, invariant with stretch factor
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Stretch parallel to -axis, invariant with stretch factor
Stretch parallel to -axis, invariant with stretch factor
Stretch parallel to
-axis,
invariant with stretch factor
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Part 3: Enlargement, Stretch and Shear
Enlargement about origin with scale factor
Enlargement about point
with scale factor
Stretch parallel to -axis, invariant with stretch factor
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Stretch parallel to -axis, invariant with stretch factor
Stretch parallel to -axis, invariant with stretch factor
Stretch parallel to
-axis,
invariant with stretch factor
Shear parallel to -axis, invariant with shear factor
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Part 3: Enlargement, Stretch and Shear Enlargement about origin with scale factor
Enlargement about point
with scale factor
Stretch parallel to -axis, invariant with stretch factor
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Stretch parallel to -axis, invariant with stretch factor
Stretch parallel to -axis, invariant with stretch factor
Stretch parallel to
-axis,
invariant with stretch factor
Shear parallel to -axis, invariant with shear factor
Shear parallel to
-axis,
invariant with shear factor
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Part 3: Enlargement, Stretch and Shear Enlargement about origin with scale factor
Enlargement about point
with scale factor
Stretch parallel to -axis, invariant with stretch factor
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Stretch parallel to -axis, invariant with stretch factor
Stretch parallel to -axis, invariant with stretch factor
Stretch parallel to
-axis,
invariant with stretch factor
Shear parallel to -axis, invariant with shear factor
Shear parallel to
-axis,
invariant with shear factor
Shear parallel to
-axis,
invariant with shear factor
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Part 3: Enlargement, Stretch and Shear Enlargement about origin with scale factor
Enlargement about point
with scale factor
Stretch parallel to -axis, invariant with stretch factor
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Stretch parallel to -axis, invariant with stretch factor
Stretch parallel to -axis, invariant with stretch factor
Stretch parallel to
-axis,
invariant with stretch factor
Shear parallel to -axis, invariant with shear factor
Shear parallel to
-axis,
invariant with shear factor
Shear parallel to
-axis,
invariant with shear factor
Shear parallel to
-axis,
invariant with shear factor
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Enlargement about origin with scale factor
The matrix of this transformation is given by
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Enlargement about origin with scale factor
The matrix of this transformation is given by
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Enlargement about origin with scale factor
The matrix of this transformation is given by
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Enlargement about origin
with scale factor
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Enlargement about origin with scale factor
The matrix of this transformation is given by
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Enlargement about origin
with scale factor
We multiply both x and y-coor. by
.
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Example 3.1
Find the enlargement about the origin with scale factor 2 of the
triangle having the coordinates
,
and C
.
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Example 3.1
Find the enlargement about the origin with scale factor 2 of the
triangle having the coordinates
,
and C
.
Multiply x and y coor by 2
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Multiply x and y coor. by 2
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Enlargement about point
with scale factor
We must change the point relative to the origin so that the centre is the
origin.
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Enlargement about point
with scale factor
We must change the point relative to the origin so that the centre is the
origin.
We subtract from to get Change
to origin .
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Enlargement about point
with scale factor
We must change the point relative to the origin so that the centre is the
origin.
We subtract from to get Change
to origin .
Then we apply the rule to enlargement about origin
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Then we apply the rule to enlargement about origin.
Enlargement about origin
with scale factor
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Enlargement about point
with scale factor
We must change the point relative to the origin so that the centre is the
origin.
We subtract from to get Change
to origin .
Then we apply the rule to enlargement about origin
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Then we apply the rule to enlargement about origin.
Enlargement about origin
with scale factor
Then we add to the resulting coordinates to return to the point
Change
to point
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Enlargement about point
with scale factor
We must change the point relative to the origin so that the centre is the
origin.
We subtract from to get Change
to origin .
Then we apply the rule to enlargement about origin
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Then we apply the rule to enlargement about origin.
Enlargement about origin
with scale factor
Then we add to the resulting coordinates to return to the point
Change
to point
subtract
multiply both coor.
coor. by
add
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Example 3.2 Find the enlargement about the point
with scale factor 2 of the
triangle having the coordinates , and C .
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Example 3.2 Find the enlargement about the point
with scale factor 2 of the
triangle having the coordinates , and C .
subtract Multiply x and y coor by 2 add
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subtract
Multiply x and y coor. by 2 add
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Figure for Enlargement for Examples 3.1 and 3.2
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Stretch parallel to
-axis,
invariant with stretch factor
The matrix of this transformation is given by
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Tricks in Transformations c-
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Stretch parallel to
-axis,
invariant with stretch factor
The matrix of this transformation is given by
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Tricks in Transformations c-
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Stretch parallel to
-axis,
invariant with stretch factor
The matrix of this transformation is given by
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Stretch along x-axis
x=0 invariant with stretch factor
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Stretch parallel to
-axis,
invariant with stretch factor
The matrix of this transformation is given by
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Stretch along x-axis
x=0 invariant with stretch factor
We multiply x-coor. by and y-coor. is unchanged.
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Example 3.3
Find the stretch parallel to -axis, invariant, stretch factor 2
of the triangle having the coordinates , and C .
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Tricks in Transformations c-
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Example 3.3
Find the stretch parallel to -axis, invariant, stretch factor 2
of the triangle having the coordinates , and C .
Multiply x coor. by 2
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Stretch parallel to
-axis,
invariant with stretch factor
We must change the point relative to the y-axis, .
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Stretch parallel to
-axis,
invariant with stretch factor
We must change the point relative to the y-axis, .
We subtract
from the x-coordinate to get
Change
to line
.
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Stretch parallel to
-axis,
invariant with stretch factor
We must change the point relative to the y-axis, .
We subtract
from the x-coordinate to get
Change
to line
.
Then we apply the rule to stretch with as invariant line.
Stretch
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as invariant line
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Stretch parallel to
-axis,
invariant with stretch factor
We must change the point relative to the y-axis, .
We subtract
from the x-coordinate to get
Change
to line
.
Then we apply the rule to stretch with as invariant line.
Stretch
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as invariant line
Then we add to the resulting x-coordinate to return to the line
Change
to line
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Stretch parallel to
-axis,
invariant with stretch factor
We must change the point relative to the y-axis, .
We subtract
from the x-coordinate to get
Change
to line
.
Then we apply the rule to stretch with as invariant line.
Stretch
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as invariant line
Then we add to the resulting x-coordinate to return to the line
Change
to line
subtract
from
coor.
multiply
coor.
by
add
to
coor.
Tricks in Transformations c-
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Example 3.4
Find the stretch parallel to -axis, invariant, stretch factor
2 of the triangle having the coordinates , and
C .
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Example 3.4
Find the stretch parallel to -axis, invariant, stretch factor
2 of the triangle having the coordinates , and
C .
subtract -1 from x-coor. (+1) Multiply x coor. by 2 add -1 to x-coor. (-1)
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Figure for Stretch parallel to
-axis for Examples 3.3 and 3.4
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Stretch parallel to
-axis,
invariant with stretch factor
The matrix of this transformation is given by
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Stretch parallel to
-axis,
invariant with stretch factor
The matrix of this transformation is given by
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Stretch parallel to
-axis,
invariant with stretch factor
The matrix of this transformation is given by
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Stretch along x-axis
y=0 invariant with stretch factor
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Stretch parallel to
-axis,
invariant with stretch factor
The matrix of this transformation is given by
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Stretch along x-axis
y=0 invariant with stretch factor
We multiply y-coor. by
and x-coor. is unchanged.
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Example 3.5
Find the stretch parallel to -axis, invariant, stretch factor 2
of the triangle having the coordinates , and C .
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Example 3.5
Find the stretch parallel to -axis, invariant, stretch factor 2
of the triangle having the coordinates , and C .
Multiply y coor. by 2
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Stretch parallel to
-axis,
invariant with stretch factor
We must change the point relative to the x-axis, .
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Stretch parallel to
-axis,
invariant with stretch factor
We must change the point relative to the x-axis, .
We subtract
from the y-coordinate to get
Change
to line
.
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Stretch parallel to
-axis,
invariant with stretch factor
We must change the point relative to the x-axis, .
We subtract
from the y-coordinate to get
Change
to line
.
Then we apply the rule to stretch with
as invariant line.
Stretch
as invariant line
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as invariant line
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Stretch parallel to
-axis,
invariant with stretch factor
We must change the point relative to the x-axis, .
We subtract
from the y-coordinate to get
Change
to line
.
Then we apply the rule to stretch with
as invariant line.
Stretch
as invariant line
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as invariant line
Then we add to the resulting y-coordinate to return to the line
Change
to line
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Stretch parallel to
-axis,
invariant with stretch factor
We must change the point relative to the x-axis, .
We subtract from the y-coordinate to get Change
to line .
Then we apply the rule to stretch with
as invariant line.
Stretch
as invariant line
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as invariant line
Then we add to the resulting y-coordinate to return to the line
Change
to line
subtract
from coor.
multiply coor.
by
add
to
coor.
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Example 3.6
Find the stretch parallel to -axis, invariant, stretch factor 2
of the triangle having the coordinates , and C .
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Example 3.6
Find the stretch parallel to -axis, invariant, stretch factor 2
of the triangle having the coordinates , and C .
subtract 1 from y-coor. (-1) Multiply y coor. by 2 add 1 to y-coor. (+1)
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Figure for Stretch parallel to
-axis for Examples 3.5 and 3.6
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Shear parallel to
-axis,
invariant with shear factor
The matrix of this transformation is given by
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Shear parallel to
-axis,
invariant with shear factor
The matrix of this transformation is given by
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Shear parallel to
-axis,
invariant with shear factor
The matrix of this transformation is given by
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Shear along x-axis
x=0 invariant with shear factor
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Shear parallel to
-axis,
invariant with shear factor
The matrix of this transformation is given by
-
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Shear along x-axis
x=0 invariant with shear factor
The x-coor. is unchanged.
We multiply x-coor. by and add the result to y-coor.to get a new y-coor.
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Example 3.7
Find the shear parallel to -axis, invariant, shear factor 2 of
the triangle having the coordinates , and C .
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Example 3.7
Find the shear parallel to -axis, invariant, shear factor 2 of
the triangle having the coordinates , and C .
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Shear parallel to
-axis,
invariant with shear factor
We must change the point relative to the y-axis, .
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Shear parallel to
-axis,
invariant with shear factor
We must change the point relative to the y-axis, .
We subtract from the x-coordinate to get Change
to line .
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Shear parallel to
-axis,
invariant with shear factor
We must change the point relative to the y-axis, .
We subtract from the x-coordinate to get Change
to line .
Then we apply the rule to shear with as invariant line.
shear
as invariant line
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Shear parallel to
-axis,
invariant with shear factor
We must change the point relative to the y-axis, .
We subtract from the x-coordinate to get Change
to line .
Then we apply the rule to shear with as invariant line.
shear
as invariant line
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Then we add
to the resulting x-coordinate to return to the line
Change
to line
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Shear parallel to
-axis,
invariant with shear factor
We must change the point relative to the y-axis, .
We subtract from the x-coordinate to get Change
to line .
Then we apply the rule to shear with as invariant line.
shear
as invariant line
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Then we add
to the resulting x-coordinate to return to the line
Change
to line
subtract
from
coor.
multiply coor. by k
and add to y coor.
add
to
coor.
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Example 3.8
Find the shear parallel to -axis, invariant, shear factor 2
of the triangle having the coordinates , and C .
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Example 3.8
Find the shear parallel to -axis, invariant, shear factor 2
of the triangle having the coordinates , and C .
subtract -1 from x-coor. (+1) Multiply x coor. by 2 + y coor. add -1 to x-coor. (-1)
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Figure for Shear parallel to
-axis for Examples 3.7 and 3.8
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Shear parallel to
-axis,
invariant with shear factor
The matrix of this transformation is given by
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Shear parallel to
-axis,
invariant with shear factor
The matrix of this transformation is given by
-
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Shear parallel to
-axis,
invariant with shear factor
The matrix of this transformation is given by
-
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shear along x-axis
y=0 invariant with shear factor
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Shear parallel to
-axis,
invariant with shear factor
The matrix of this transformation is given by
-
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shear along x-axis
y=0 invariant with shear factor
The y-coor. is unchanged.
We multiply y-coor. by and add the result to x-coor.to get a new x-coor.
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Example 3.9
Find the shear parallel to -axis, invariant, shear factor 2 of
the triangle having the coordinates , and C .
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Example 3.9
Find the shear parallel to -axis, invariant, shear factor 2 of
the triangle having the coordinates , and C .
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Shear parallel to
-axis,
invariant with shear factor
We must change the point relative to the x-axis, .
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Shear parallel to
-axis,
invariant with shear factor
We must change the point relative to the x-axis, .
We subtract from the y-coordinate to get Change
to line
.
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Shear parallel to
-axis,
invariant with shear factor
We must change the point relative to the x-axis, .
We subtract from the y-coordinate to get Change
to line
.
Then we apply the rule to shear with
as invariant line.
shear
as invariant line
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Shear parallel to
-axis,
invariant with shear factor
We must change the point relative to the x-axis, .
We subtract from the y-coordinate to get Change
to line
.
Then we apply the rule to shear with
as invariant line.
shear
as invariant line
Then we add to the resulting y-coordinate to return to the line
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Then we add
to the resulting y-coordinate to return to the line
Change
to line
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Shear parallel to
-axis,
invariant with shear factor
We must change the point relative to the x-axis, .
We subtract from the y-coordinate to get Change
to line
.
Then we apply the rule to shear with
as invariant line.
shear
as invariant line
Then we add to the resulting y-coordinate to return to the line
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Then we add
to the resulting y coordinate to return to the line
Change
to line
subtract
from coor.
multiply coor.
by and add to x coor.
add
to
coor.
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Example 3.10
Find the shear parallel to -axis, invariant, shear factor 2 of
the triangle having the coordinates , and C .
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Example 3.10
Find the shear parallel to -axis, invariant, shear factor 2 of
the triangle having the coordinates , and C .
subtract 1 from y-coor. (-1) Multiply y coor. by 2
x coor. add 1 to y-coor. (+1)
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Figure for Shear parallel to
-axis for Examples 3.9 and 3.10
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