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Further Pure 1
Lesson 2 –
Transformations
Transformations
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 2 × 2 matrices can be used to describe
transformations in a 2-d plane.
 Before we look at this we are going to look at
particular transformations in the 2D plane.
 A transformation is a rule which moves points
about on a plane.
 Every transformation can be described as a
multiple of x plus a multiple of y.
Transformations
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 Lets look at a point A(-2,3) and map it to the co-
ordinate (2x+3y,3x-y)
 This gives us the
co-ordinate
(2×-2 + 3×3, 3×-2–3)
=(5,-9)
 Where would the
co-ordinate (2,1) map
to?
(7,5)
(-2,3)
(2,1)
(5,-9)
Transformations
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 Take the transformation reflecting an object in the y-axis.
 The black rectangle is the object and the orange one is the
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image.
What has happened to the co-ordinates in the reflection?
Lets look at one specific co-ordinate, (2,1).
Under the reflection the coordinate becomes (-2,1)
You can probably notice that
there is a general rule for all the
co-ordinates.
(-2,1)
(2,1)
For each co-ordinate the x
becomes negative and the y
stays the same.
Lets use the general co-ordinate
(x,y) and let them map to (x`,y`).
Reflection in y-axis
 We can see that x
 Or x` = -x
-x & y
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y.
y` = y
 So we can now write these equations as a pair of
simultaneous equations as multiples of x and y.
x` = -1x + 0y
y` = 0x + 1y
 Finally we can summarise the
equations co-efficient’s by using
matrix notation.
(-2,1)
 1 0


 0 1
(2,1)
Reflection in x-axis
 We can see that x
 Or x` = x
x&y
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-y.
y` = -y
 So we can now write these equations as a pair of
simultaneous equations as multiples of x and y.
x` = 1x + 0y
y` = 0x + -1y
 Finally we can summarise the
equations co-efficient’s by using
matrix notation.
1 0 


 0 - 1
(2,1)
(2,-1)
Reflection in y = x
 We can see that x
 Or x` = y
y&y
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x.
y` = x
 So we can now write these equations as a pair of
simultaneous equations as multiples of x and y.
x` = 0x + 1y
y` = 1x + 0y
 Finally we can summarise the
(1,2)
equations co-efficient’s by using
matrix notation.
 0 1


 1 0
(2,1)
Reflection in y = -x
 We can see that x
 Or x` = -y
-x & y
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y.
y` = -x
 So we can now write these equations as a pair of
simultaneous equations as multiples of x and y.
x` = 0x + -1y
y` = -1x + 0y
 Finally we can summarise the
equations co-efficient’s by using
matrix notation.
 0 - 1


-1 0 
(-1,-2)
(2,1)
Enlargement SF 2, centre (0,0)
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 We can see that x
 Or x` = 2x
2x & y
2y.
y` = 2y
 So we can now write these equations as a pair of
simultaneous equations as multiples of x and y.
x` = 2x + 0y
y` = 0x + 2y
 Finally we can summarise the
equations co-efficient’s by using
matrix notation.
2 0


0 2
(2,4)
(1,2)
Two way stretch
 We can see that x
 Or x` = 2x
2x & y
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3y.
y` = 3y
 So we can now write these equations as a pair of
simultaneous equations as multiples of x and y.
x` = 2x + 0y
y` = 0x + 3y
 Finally we can summarise the
equations co-efficient’s by using
matrix notation.
2 0


0 3
 This is a stretch factor 2 for x
and factor 3 for y.
(4,3)
(2,1)
Enlargements
 Enlargement
SF k
k 0


0 k 
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 Two way stretch
 Factor a for x
 Factor b for y
a 0


0 b
Rotation 90o anti-clockwise
 We can see that x
 Or x` = -y
-y & y
x.
y` = x
 So we can now write these equations as a pair of
simultaneous equations as multiples of x and y.
x` = 0x – 1y
(-2,4)
y` = 1x + 0y
 Finally we can summarise the
equations co-efficient’s by using
matrix notation.
 0 - 1


1 0 
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(4,2)
Rotation 90o clockwise
 We can see that x
 Or x` = y
y&y
-x.
y` = -x
 So we can now write these equations as a pair of
simultaneous equations as multiples of x and y.
x` = 0x + 1y
y` = -1x + 0y
 Finally we can summarise the
equations co-efficient’s by using
matrix notation.
 0 1


- 1 0
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(4,2)
(2,-4)
Rotation 180o
 We can see that x
 Or x` = -x
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-x & y
-y.
y` = -y
 So we can now write these equations as a pair of
simultaneous equations as multiples of x and y.
x` = -1x + 0y
y` = 0x – 1y
 Finally we can summarise the
equations co-efficient’s by using
matrix notation.
-1 0 


 0 - 1
(-4,-2)
(4,2)
Rotation through θ anti-clockwise.
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 We are going to think about this example in a slightly different
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way.
The diagram shows the points I(1,0) and J(0,1) and there
images after a rotation through θ anti-clockwise.
You can see OI = OJ = OI` = OJ`
From the diagram we can see that
b
J(0,1) I`(a,b)
cos θ = a/1  a = cos θ
J`(-b,a)
sin θ = b/1  b = sin θ
a
1
1
Therefore I` is (cos θ, sin θ) and
b
J` is (-sin θ, cos θ)
The transformation matrix is
a
I(1,0)
 cosθ - sinθ 


 sinθ cosθ 
Rotation through θ clockwise.
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 What would be the matrix for a 90o rotation
clockwise.
 cosθ sinθ 


 - sinθ cosθ 
Transformations - Shears
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 For the next example you need to understand the concept of a
shear.
 Here is an example of a shear parallel to the x-axis factor 2.
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Each point moves parallel
to the x-axis.
Each point is moved twice
its distance from the x-axis.
Points above the x-axis
move right.
Points below the x-axis
move left.
 You can see that the point (2,1)
moves to (2 + 2 × 1,1) = (4,1)
 A shear parallel to the y-axis
factor 3 would move every point
3 times its distance from y
parallel to the y-axis.
Shear parallel to x-axis factor 2
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 We can see that x
 Or x` = x + 2y
x + 2y & y
y.
y` = y
 So we can now write these equations as a pair of
simultaneous equations as multiples of x and y.
x` = 1x + 2y
y` = 0x + 1y
 Finally we can summarise the
equations co-efficient’s by using
matrix notation.
 1 2


 0 1
(2,1) (4,1)
Shear parallel to y-axis factor 2
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 We can see that x
 Or x` = x
x&y
y + 2x .
y` = 2x + y
 So we can now write these equations as a pair of
simultaneous equations as multiples of x and y.
x` = 1x + 0y
y` = 2x + 1y
 Finally we can summarise the
equations co-efficient’s by using
matrix notation.
 1 0


 2 1
(2,5)
(2,1)
Two way shear factor 2
 We can see that x
 Or x` = x + 2y
x + 2y &
y
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y + 2x.
y` = 2x + y
 So we can now write these equations as a pair of
simultaneous equations as multiples of x and y.
x` = 1x + 2y
y` = 2x + 1y
 Finally we can summarise the
equations co-efficient’s by using
matrix notation.
 1 2


 2 1
(4,5)
(2,1)
Using multiplication with
transformations
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 Lets go back to the first transformation that we looked at.
 We know that the matrix for reflecting in the y-axis is
  1 0 1 2 2 1    1  2  2  1


  

1
3
3
 0 1 1 1 3 3   1
 Now lets write down the co-
ordinates of the object as a
matrix.
 What happens if we multiply the
two matrices together.
 The multiplication performs the
transformation and the new
matrix is the co-ordinates of the
image.
Rotation 180o
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 What happens if you rotate 90o cw, twice.
 0  1 0  1   1 0 


  

 1 0  1 0   0  1
 What happens if you reflect in x then in y.
 1 0   1 0    1 0 


  

 0  1 0 1   0  1
 You actually get the same transformation as rotating
through 180o.
1 0 


 0  1
 This leads us nicely in to multiple transformations.
Composition of transformations
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 Notation:
 A single bold italic letter such as T is often used to
represent a transformation.
 A bold upright T is used to represent a matrix
itself.
 If you have a point P with position vector p
 The image of p can be denoted
P` = p` = T(P)
 If you transform p by a transformation X then by
a transformation Y the result would be:
Y(X(p)) = YX(p)