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 1 3 2 –2 1 3 + = –3 –3 –3 1 1 1 –2 0 –1 –1 2 4 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).

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 0 2 3 –1 –2 3 2 –2 –3 0 = 0 2 – – 2 – –2 0 4 Multiply. The new coordinates are A (0, 2), B ( , ), C (2, – ), D (– , –2), and E (– , 0). 4 3 2

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

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

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

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