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. 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 ABC. The coordinates of the vertices of the image are A (–2, –1), B (0, 2), C (–1, 4). Vertices ofTranslationVertices of the Triangle Matrixthe 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 Translating a Figure

5. The figure in the diagram is to be reduced by a factor of (Dilation). Find the coordinates of the vertices of the reduced figure. 2323 Write a matrix to represent the coordinates of the vertices. A B C D E A B C D E 2323 0 2 3 –1 –2 3 2 –2 –3 0 = 0 2 – – 2 – –2 0 4343 4343 4343 4343 2323 Multiply. The new coordinates are A (0, 2), B (, ), C (2, – ), D (–, –2), and E (–, 0). 4343 4343 4343 2323 4343 Real World Example

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

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

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

10. Homework Pg 190 # 2-22 even