2. 2
Transformation
• Definition of transformation: To
moving a point or object in a plane.
• Transformation T on a plane is just
‘mapping' each point P on the plane
to P' on the same plane.
• Point P' is called the image or the
mapping of point P.
3. 3
Inverse of Transformation
To determine the image of a curve under
a transformation that expressed in
matrix form, we can use the inverse
transformation.
4. 4
Example of Question
The image of the line x – 2y + 5 = 0
under the transformation that
expressed by matrix
is….
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5. 5
Discussion
A(x,y) A’(x’, y’)
Remind: A = BX then X = B-1.A
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7. 7
x = 3x’ – y’ and y = -2x’ + y’
Substitute the result in:
x – 2y + 5 = 0
3x’ – y’ – 2(-2x’ + y’) + 5 = 0
3x’ – y’ + 4x’ – 2y’ + 5 = 0
7x’ – 3y’ + 5 = 0
So we get the image of the line above
is:
7x – 3y + 5 = 0
8. 8
Composition of Transformation 1
If T1 is a transformation from point A(x,y) to
point A’(x’,y’) followed by transformation T2 is
transformation from A’(x’,y’) to point A”(x”,y”)
then the two successive transformations are
called composition of transformation with the
symbol T2 o T1
9. 9
Composition of Transformation
using Matrix
If T1 is expressed in matrix form
and T2 with matrix
then the two successive transformations . First
from T1 then followed by T2 can be written as
follows T2 o T1 =
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10. 10
Example 1
The matrix corresponding to the
dilatation with the center O(0,0)
and scale factor 3, then followed
by a reflection across the line y=x,
is…
11. 11
Discussion
M1= Matrix with dilatation scale 3
which is
M2 = Matrix reflection by line
y = x which is
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12. 12
Matrix that relevant with
M1 then followed by M2
is M2 o M1 =
=
So, we get the matrix:
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13. 13
Example 2
The image of triangle ABC, with A(2,1),
B(6,1), C(5,3), under the reflection across
the Y-axis followed by a rotation of (0,
π), is...
15. 15
Example 3
T1 is the transformation
corresponding to the matrix
and T2 is the transformation
corresponding to the matrix
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16. 16
Image point A(m,n) by transformation
T1 then followed by T2 is A’(-9,7).
Value of m - 2n is….
18. 18
T2 o T1 =
A(m,n) A’(-9,7)
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19. 19
We get : -9 = m + n and 7 =
m
Substitute m = 7 into
m + n = -9 Þ 7 + n = -9
n = -16
Then we get the result:
m – 2n = 7 + 32 = 39
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20. 20
Example 4
If the point (a,b) is reflected across the Y-
axis, then followed by a transformation
corresponding to the matrix
and resulting the point (1,-8). Then, the
value of a + b =….
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21. 21
Discussion
The matrix of reflection across the Y-
axis : T1 =
T2 =
T2 o T1 =
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23. 23
Example 5
The equation of the image curve of
y = x2 - 3x + 2 under the reflection
across the x-axis and followed by a
dilatation with center O and scale
factor ⅓ is…
25. 25
from x” = ⅓x and y” = -⅓y
We get x = 3x” and y = -3y”
Then, substitute into y = x2 – 3x + 2
-3y” = (3x”)2 – 3(3x”) + 2
-3y” = 9(x”)2 – 9x” + 2
So we get the image of the equation
is: y = -3x2 + 3x - ⅔
26. 26
Example 6
The equation of the image curve obtained
by reflecting a curve across the X-axis,
And then followed by translation
is y = x2 – 2. The original equation of the
curve is…
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2
