TRANSFORMATIONS
Reflection In The X-Axis
Reflection In The x-Axis
Consider what happens to the unit square.
1
1
OA =
.
OA′=
also
0
0
0
0
O𝐵 =
.
OB′=
1
−1
1
0
Combining these gives
, which
0 −1
represents reflection in the x-axis.
Example:
Find the image of triangle A(1, 2), B(2, 1),
C(1, 1) after reflection in the x –axis.
A B C
A’ B’ C’
1
0 1 2 1
1
2
1
=
0 −1 2 1 1
−2 −1 −1
The image is given by the coordinates:
A’ (1, -2), B’(2, -1), C’(1, -1)
y
2
1 B
A
-2
-1
1
0
-1
-2
B'
2
x
Reflection In The y -Axis
Consider what happens to the unit square.
1
−1
OA =
.
OA′=
also
0
0
0
0
OB =
.
OB′=
1
1
−1 0
Combining these gives
, which
0 1
represents reflection in the y -axis.
Example:
Find the image of triangle A(1, 2), B(2, 1),
C(1, 1) after reflection in the y –axis.
A B C
A’ B’ C’
−1 0 1 2 1
−1 −2 −1
=
0 1 2 1 1
2
1
1
The image is given by the coordinates:
A’ (-1, 2), B’(-2, 1), C’(-1, 1)
y
2
1 B
A'
-2
-1
A
1
0
-1
-2
B'
2
x
Reflection In The Line y = x
Consider what happens to the unit square.
1
0
OA =
.
OA′=
also
0
1
1
0
OB =
.
OB′=
0
1
0 1
Combining these gives
, which
1 0
represents reflection in the line y = x.
2
1B
-2
-1
0
x
-1
y=
Example:
Find the image of triangle A(1, 2), B(2, 1),
C(1, 1) after reflection in the line y = x.
A B C
A’ B’ C’
0 1 1 2 1
2 1 1
=
1 0 2 1 1
1 2 1
The image is given by the coordinates:
A’ (2, 1), B’(1, 2), C’(1, 1)
y
-2
A'
A
1B'
2
x
Reflection In The Line y = -x
Consider what happens to the unit square.
1
0
OA =
.
OA′=
also
0
−1
−1
0
OB =
.
OB′=
0
1
0 −1
Combining these gives
, which
−1
0
represents reflection in the line y = x.
y
y=
2
-x
1 B
B'
Example:
Find the image of triangle A(1, 2), B(2, 1),
C(1, 1) after reflection in the line y = x.
A B C
A’ B’ C’
0 −1 1 2 1
−2 −1 −1
=
−1
0 2 1 1
−1 −2 −1
The image is given by the coordinates:
A’ (-2, -1), B’(-1, -2), C’(-1, -1)
-2
-1
A
1
0
-1
-2
A'
2
x
Reflection In The Origin
Consider what happens to the unit square.
1
−1
OA =
.
OA′=
also
0
0
0
0
OB =
.
OB′=
1
−1
−1 0
Combining these gives
, which
0 −1
represents reflection in the origin
Example:
Find the image of triangle A(1, 2), B(2, 1),
C(1, 1) after reflection in the line y = x.
A B C
A’ B’ C’
0 1 1 2 1
2 1 1
=
1 0 2 1 1
1 2 1
The image is given by the coordinates:
A’ (2, 1), B’(1, 2), C’(1, 1)
y
2
1
B
A'
-2
-1
A
0
-1
-2
1
B'
2
x
Rotation 90o (anti-clockwise)
y
Consider what happens to the unit square.
1
OA =
.
0
OB =
0
.
1
2
0
OA′=
also
1
OB′=
A'1B
−1
0
0 −1
Combining these gives
, which
1
0
represents a rotation through 90o clockwise.
B'
-2
-1
A
0
-1
Generally, a rotation about the origin, O,
through an angle, , is given by the matrix:
-2
cos 𝜃
sin 𝜃
− sin 𝜃
cos 𝜃
1
2
x
Example:
Find the image (ABC) of triangle
ABC under rotation a rotation of 90o anticlockwise, with A (2, 3) B (3, 3) and C
(2, 1)
Rotation 90oAnti-clockwise Centre Origin
y
4
Solution
𝟎
Transformation matrix is
𝟏
𝑹𝟗𝟎
𝟎 −𝟏
𝟏
𝟎
−𝟏
𝟎
A B C
A
B
C
𝟐 𝟑 𝟐
−𝟑 −𝟑 −𝟏
=
𝟑 𝟑 𝟏
𝟐
𝟑
𝟐
The image is given by the coordinates:
A(-3, 2), B(-3, 3), C(-1, 2) which are
as plotted in the diagram.
Generally, for a rotation about the origin:
cos 𝜃 − sin 𝜃
𝑅𝜃 =
sin 𝜃
cos 𝜃
B'
3
A'
C' 2
A
1
-4
-3
-2
-1
0
=1
-2
-3
B
C
1
2
3
4
x
Rotations
By convention an anti-clockwise rotation is considered positive, and a clockwise rotation is
negative.
To describe a rotation requires three
y
pieces of information;
Q (0, 1) P' (0, 1)
(a) The angle of rotation;
(b) The direction of rotation;
(c) The centre of rotation.
Transformation Matrix
Find the matrixoof transformation for
a rotation of 90 .
With centre, O, rotate OP through 90o this
gives
the point P’ (0, 1) as the image of P.
With centre, O, rotate OQ through 90o
this gives the point Q’ (-1, 0) as the image of Q.
Q' (-1, 0)
0
(0, -1)
x
P (1, 0)
𝑎 𝑏
be the matrix for the transformation.
𝑐 𝑑
𝑎
0
0
𝑎 𝑏 1
=
∴
=
,  a = 0, c = 1
𝑐
1
1
𝑐 𝑑 0
Similarly
Let
𝑎 𝑏
be the matrix for the transformation.
𝑐 𝑑
−1
𝑎 𝑏 0
=
0
𝑐 𝑑 1
−1
𝑏
∴
=
,  b = -1, d = 0
0
𝑑
0 −1
The required matrix is therefore
.
1
0
Generally, a rotation about the origin, O,
through an angle, , is given by the matrix:
cos 𝜃 − sin 𝜃
sin 𝜃
cos 𝜃
y
Q (0, 1) P' (0, 1)
Let
Q' (-1, 0)
0
(0, -1)
x
P (1, 0)