tricks in transformations

7/29/2019 Tricks in Transformations

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Tricks in Transformations c- [email protected]

Dr D K R Babajee

[email protected]

Mauritius

Tricks in Transformations c-

[email protected] p.1/66


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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 y-axis




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Part 1: Reflection


Reflection in the y-axis Reflection in the vertical line




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Part 1: Reflection



Reflection in the line y=x




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Part 1: Reflection



Reflection in the line y=x Reflection in the line



fl


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Part 1: Reflection




Reflection in the line y=-x



P 1 R fl i


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Part 1: Reflection




Reflection in the line y=-x Reflection in the line



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

Finding the mirror line (horizontal or vertical line)



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



Reflection in the a is ( 0)


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Reflection in the x axis (y 0)


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Reflection

in x-axis



Reflection in the x axis (y 0)


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Reflection

in x-axis

-coordinate is unchanged and we reverse the sign ofy-coordinate.



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.



Example 1 1


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Example 1.1


and C in the x-axis.

reverse sign of coor.





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We must change the point relative to the x-axis .





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We subtract from the y-coordinate to get Change

to line .





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to line .

Then we apply the rule to reflection in x-axis.

Reflection

in x-axis





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to line .


Reflection

in x-axis

Then we add to the resulting y-coordinate to return to the line

Change

to line





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to line .


Reflection

in x-axis


Change

to line

subtract

from

reverse sign

of

add

to



Example 1.2


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Example 1.2

Find the reflection of the triangle having the coordinates

,

and C in the line .



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.



Reflection in the y-axis (x=0)


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y ( )






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y ( )






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y ( )


Reflection

in y-axis





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Reflection

in y-axis

we reverse the sign of x-coordinate and -coordinate isunchanged.



Example 1.3


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and C in the y-axis.



Example 1.3


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and C in the y-axis.

reverse sign of coor.





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We must change the point relative to the y-axis, .





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We subtract from the x-coordinate to get Change

to line .





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to line .

Then we apply the rule to reflection in y-axis.

Reflection

in y-axis





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to line .


Reflection

in y-axis

Then we add to the resulting x-coordinate to return to the line

Change

to line





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to line .


Reflection

in y-axis


Change

to line

subtract

from

reverse sign

of

add

to



Example 1.4


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Find the reflection of the triangle having the coordinates , and

C in the line .



Example 1.4


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C in the line .

subtract

from

coor. (+1) reverse sign of

coor. add -1 to

coor. (-1)



Figure for Reflection for Examples 1.1 to 1.4


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Reflection

in y=x





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Reflection

in y=x

Interchange the coordinates of and .



Example 1.5


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and C in the line .



Example 1.5


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and C in the line .

Interchange



Reflection in the line


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We must change the point relative to the line .





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to line .





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to line .

Then we apply the rule to reflection in the line .

Reflection

in line y=x





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to line .


Reflection

in line y=x


Change

to line





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to line .


Reflection

in line y=x


Change

to line

subtract

from

interchange

x and y-c

add

to new y coor.



Example 1.6


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C in the line .



Example 1.6


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C in the line .

subtract

from coor. interchange add 1 to coor.





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Th i f hi f i i i b


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Th t i f thi t f ti i i b


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Reflection

in y=-x




Th t i f thi t f ti i i b


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Reflection

in y=-x

Interchange the coordinates of

and

and reversetheir signs.



Example 1.7


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and C in the line .



Example 1.7

Fi d h fl i f h i l h i h di


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and C in the line .

Interchange Reverse signs of both coor.




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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to line .






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to line .


Reflection

in line y=-x






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to line .


Reflection

in line y=-x


Change

to line






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to line .


Reflection

in line y=-x


Change

to line

subtract

from

interchange and reverse signs

add

to new y coor.



Example 1.8

Find the reflection of the triangle having the coordinates and


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C in the line .



Example 1.8

Find the reflection of the triangle having the coordinates and


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C in the line .

subtract

from

coor. (+1) interchange and reverse signs add

to

coor. (-1)



Figure for Reflection for Examples 1.5 to 1.8


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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.




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.



Example 1.9

Describe the transformation when the triangle

is mapped on triangle

:


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g pp g



Example 1.9



:


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Example 1.9



:


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We observe that y-coor. are unchanged and so it is a reflection in the line .



Example 1.9



:


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To find , we find the x-coor. of the midpoint of the object and image.



Example 1.9



:


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Example 1.9



:


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.



Example 1.10



:


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Example 1.10



:


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Example 1.10



:


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We observe that x-coor. are unchanged and so it is a reflection in the line .



Example 1.10



:


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To find , we find the y-coor. of the midpoint of the object and image.



Example 1.10



:


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Example 1.10



:


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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



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Rotation about the point



Part 2: Rotation



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Anticlockwise Rotation about the origin



Part 2: Rotation



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Anticlockwise Rotation about the point



Part 2: Rotation



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Clockwise Rotation about the origin



Part 2: Rotation


R i b h i


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Clockwise Rotation about the point






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Rotation

about origin






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Rotation

about origin

We reverse the signs of both x- and y-coordinates.



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.



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.




We must change the point relative to the origin so that the centre is the

origin


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origin.





origin


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origin.

We subtract from to get Change

to origin .





origin


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origin.


to origin .

Then we apply the rule to

rotation about origin.

Rotation

about origin





origin.


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origin.


to origin .



Rotation

about origin

Then we add to the resulting coordinates to return to the point

Change

to point





origin.


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origin.


to origin .



Rotation

about origin


Change

to point

subtract

reverse signs of

x and y coor.

add



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 .



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 origin






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e at o t s t a s o at o s g e by

Anticlockwise


We reverse the sign of y-coordinate and interchangethe resulting coordinates.



Example 2.3

Find the anticlockwise rotation of the triangle having the

coordinates , and C about the origin.


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g



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



Anticlockwise about the point


origin.


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origin.

Change


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to origin .





origin.

W bt t f t tChange


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to origin .


anticlockwise rotation about origin.

anticlockwise

rotation about origin





origin.

W bt t f t tChange


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We subtract from to get g

to origin .



anticlockwise


Then we add

to the resulting coordinates to return to the point

Change

to point





origin.

We subtract from to getChange


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We subtract from to get g

to origin .



anticlockwise


Then we add


Change

to point

subtract

reverse sign of y coor.

and interchange

add

(a,b)



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







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Clockwise


We reverse the sign of x-coordinate and interchangethe resulting coordinates.



Example 2.5

Find the clockwise rotation of the triangle having the

coordinates

,

and C

about the origin.


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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



clockwise about the point


origin.


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origin.


t i i .


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gto origin





origin.


t i i .


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to origin


clockwise rotation about origin.

clockwise






origin.


to origin .


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to origin



clockwise


Then we add


Change

to point





origin.


to origin .


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to origin



clockwise


Then we add


Change

to point

subtract

reverse sign of x coor.

and interchange

add

(a,b)



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



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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Enlargement about point

with scale factor


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with scale factor

Stretch parallel to -axis, invariant with stretch factor


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with scale factor



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with scale factor



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with scale factor



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Stretch parallel to

-axis,

invariant with stretch factor






with scale factor



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Stretch parallel to

-axis,


Shear parallel to -axis, invariant with shear factor



Part 3: Enlargement, Stretch and Shear Enlargement about origin with scale factor


with scale factor



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Stretch parallel to

-axis,



Shear parallel to

-axis,

invariant with shear factor





with scale factor



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Stretch parallel to

-axis,



Shear parallel to

-axis,


Shear parallel to

-axis,






with scale factor



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Stretch parallel to

-axis,



Shear parallel to

-axis,


Shear parallel to

-axis,


Shear parallel to

-axis,







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Enlargement about origin

with scale factor






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with scale factor

We multiply both x and y-coor. by

.



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




with scale factor


origin.


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with scale factor


origin.


to origin .


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with scale factor


origin.


to origin .

Then we apply the rule to enlargement about origin


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Then we apply the rule to enlargement about origin.


with scale factor




with scale factor


origin.


to origin .



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with scale factor


Change

to point




with scale factor


origin.


to origin .



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with scale factor


Change

to point

subtract

multiply both coor.

coor. by

add



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



Figure for Enlargement for Examples 3.1 and 3.2


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Stretch parallel to

-axis,




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Stretch parallel to

-axis,




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Stretch parallel to

-axis,




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Stretch along x-axis

x=0 invariant with stretch factor



Stretch parallel to

-axis,




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x=0 invariant with stretch factor

We multiply x-coor. by and y-coor. is unchanged.



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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Example 3.3



Multiply x coor. by 2


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Stretch parallel to

-axis,




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Stretch parallel to

-axis,



We subtract

from the x-coordinate to get

Change

to line

.


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Stretch parallel to

-axis,



We subtract


Change

to line

.

Then we apply the rule to stretch with as invariant line.

Stretch


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as invariant line



Stretch parallel to

-axis,



We subtract


Change

to line

.


Stretch


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as invariant line


Change

to line



Stretch parallel to

-axis,



We subtract


Change

to line

.


Stretch


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as invariant line


Change

to line

subtract

from

coor.

multiply

coor.

by

add

to

coor.



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,




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Stretch parallel to

-axis,




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Stretch parallel to

-axis,




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y=0 invariant with stretch factor



Stretch parallel to

-axis,




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y=0 invariant with stretch factor

We multiply y-coor. by

and x-coor. is unchanged.



Example 3.5




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Example 3.5



Multiply y coor. by 2


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Stretch parallel to

-axis,


We must change the point relative to the x-axis, .


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Stretch parallel to

-axis,



We subtract

from the y-coordinate to get

Change

to line

.


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Stretch parallel to

-axis,



We subtract


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



Stretch parallel to

-axis,



We subtract


Change

to line

.


as invariant line.

Stretch

as invariant line


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as invariant line


Change

to line



Stretch parallel to

-axis,




to line .


as invariant line.

Stretch

as invariant line


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as invariant line


Change

to line

subtract

from coor.

multiply coor.

by

add

to

coor.



Example 3.6




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Example 3.6



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



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Shear parallel to

-axis,




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Shear parallel to

-axis,




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Shear parallel to

-axis,




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Shear along x-axis

x=0 invariant with shear factor



Shear parallel to

-axis,




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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.



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




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Shear parallel to

-axis,




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Shear parallel to

-axis,




to line .


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Shear parallel to

-axis,




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,




to 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



Shear parallel to

-axis,




to 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.



Example 3.8

Find the shear parallel to -axis, invariant, shear factor 2



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Example 3.8

Find the shear parallel to -axis, invariant, shear factor 2


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



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Shear parallel to

-axis,




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Shear parallel to

-axis,




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Shear parallel to

-axis,




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shear along x-axis

y=0 invariant with shear factor



Shear parallel to

-axis,




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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.



Example 3.9




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Example 3.9




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Shear parallel to

-axis,




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Shear parallel to

-axis,




to line

.


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Tricks in Transformations c- [email protected] p.64/66

Shear parallel to

-axis,




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,




to line

.


as invariant line.

shear

as invariant line



204/208

Then we add

to the resulting y-coordinate to return to the line

Change

to line


Shear parallel to

-axis,




to line

.


as invariant line.

shear

as invariant line



205/208

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.


Example 3.10




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Example 3.10



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



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