1. 4-4 Geometric Transformations with Matrices
Objectives: to represent translations and dilations w/ matrices
: to represent reflections and rotations with matrices

2. Objectives
Translations & Dilations w/ Matrices Reflections & Rotations w/ Matrices

3. Vocabulary
A change made to a figure is a transformation of the figure.
The transformed figure is the image.
The original figure is the preimage.

4. Translating a Figure
Triangle ABC has vertices A(1, –2), B(3, 1) and C(2, 3). Use a matrix to find the vertices of the image translated 3 units left and 1 unit up. Graph ABC and its image ABC.
Vertices of Translation Vertices of
the Triangle Matrix the image – =
–3 –3 –3
– –1 –
Subtract 3 from each
x-coordinate.
Add 1 to each
y-coordinate.
A B C A B C
The coordinates of the vertices of the image are
A (–2, –1), B (0, 2), C (–1, 4).

5. Real World Example
The figure in the diagram is to be reduced by a factor of . Find the coordinates of the vertices of the reduced figure. 2 3
Write a matrix to represent the coordinates of the vertices.
A B C D E A B C D E 2 3
–1 –2
–2 – =
– –
– – 4
Multiply.
The new coordinates are A (0, 2), B ( , ), C (2, – ),
D (– , –2), and E (– , 0). 4 3 2

6. Matrices for Reflections
Reflection in the y-axis
Reflection in the x-axis
Reflection in the line y = x
Reflection in the y = -x

7. Reflecting a Figure
Reflect the triangle with coordinates A(2, –1), B(3, 0), and C(4, –2) in each line. Graph triangle ABC and each image on the same coordinate plane.
a. x-axis
–1 0
0 1
– –2 =
–2 –3 –4
– –2
0 1
1 0
– –2 =
– –2
0 –1
– –2 =
0 –1 –
– –2 =
–2 –3 –4
b. y-axis
c. y = x
d. y = –x

8. Continued (continued) a. x-axis 1 0 0 –1 2 3 4 –1 0 –2 = 1 0 2 0 –1
0 –1
– –2 =
0 –1 –
– –2 =
–2 –3 –4
b. y-axis
–1 0
0 1
– –2 =
–2 –3 –4
– –2
c. y = x
1 0
0 1
– –2 =
– –2
d. y = –x

9. Matrices for Rotations
Rotation of 90˚
Rotation of 180˚
Rotation of 270˚
Rotation of 360˚

10. Rotating a Figure
Rotate the triangle from Additional Example 3 as indicated. Graph the triangle ABC and each image on the same coordinate plane.
a. 90 –
– –2 =
– –2
–2 –3 –4
– –2 = –
0 –1
– –2 =
–2 –3 –4
0 –1
– –2 =
b. 180
c. 270
d. 360

11. Homework
Pg 195 & # 1,5,10,11,13,14