4 4 Geometric Transformations with Matrices Objectives to

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

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

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.

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 A 1 – 2 B C 3 1 2 3 A B C + – 3 1 = – 2 – 1 0 2 – 1 4 Subtract 3 from each x-coordinate. Add 1 to each y-coordinate. The coordinates of the vertices of the image are A (– 2, – 1), B (0, 2), C (– 1, 4).

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

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

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 2 3 4 = 0 – 1 0 – 2 1 0 2 b. y-axis – 1 0 2 3 4 – 2 – 3 – 4 = 0 1 – 1 0 – 2 c. y = x 0 1 2 3 4 – 1 0 – 2 = 1 0 – 2 2 3 4 d. y = –x 0 – 1 2 3 4 1 0 2 = – 1 0 – 2 – 3 – 4

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

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

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 0 – 1 1 0 b. 180 – 1 0 0 – 1 2 – 1 3 4 0 – 2 = 1 2 0 3 2 4 2 – 1 3 4 0 – 2 = – 2 – 3 – 4 1 0 2 2 – 1 3 4 0 – 2 = – 1 0 – 2 – 3 – 4 = 2 – 1 c. 270 0 1 – 1 0 d. 360 1 0 0 1 2 – 1 3 4 0 – 2

Homework Pg 195 & 196 # 1, 5, 10, 11, 13, 14