4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

1) Translations and Dilations with Matrices When we move a figure without changing its size or shape, it is a translation. When we change the size of a figure to make it bigger or smaller, it is a dilation. The original figure is called the preimage and the new, transformed figure is called the image.

1) Translations and Dilations with Matrices Translation Preimage Image Moved the preimage… Size and shape did not change

1) Translations and Dilations with Matrices Dilation Preimage Image Enlarged the preimage… Size and shape did change

Example 1: Given triangle ABC with A (–4, 1), B (– 2, 5) and C (0, 2), translate the preimage 5 units right and 3 units down. Then, sketch the image. 1) Translations

Example 1: Given triangle ABC with A (–4, 1), B (– 2, 5) and C (0, 2), translate the preimage 5 units right and 3 units down. Then, sketch the image. So…all x values move right 5 units all y values move down 3 units 1) Translations

Step 1: Write the vertices of the triangle as a matrix 1) Translations Vertices of the Triangle AB C xyxy

Step 2: Write a translation matrix 1) Translations Vertices of the Triangle AB C Add 5 to each x- coordinate Subtract 3 from each y-coordinate xyxy

Step 3: Add the matrices 1) Translations Vertices of the Triangle AB C Add 5 to each x- coordinate Subtract 3 from each y-coordinate Vertices of the Image A’ B’ C’ xyxy

Step 3: Add the matrices 1) Translations Vertices of the Triangle AB C Add 5 to each x- coordinate Subtract 3 from each y-coordinate Vertices of the Image A’ B’ C’ xyxy “A prime” “B prime” “C prime”

1) Translations Image Preimage

1) Translations Example 2: Translate the image 2 units left and 1 unit up

1) Translations Example 2: Vertices of the Preimage A B C D E Subtract 2 from each x-coordinate Add 1 to each y- coordinate + = Vertices of the Image A’ B’ C’ D’ E’

1) Translations and Dilations with Matrices Example 2: Preimage Image

1) Translations and Dilations with Matrices Example 2: Preimage Image

Example 1: Given triangle ABC with A (–2,0), B (0, 4) and C (2, 1) increase the size of the triangle by a factor of 1.5. Then, sketch the image. 1) Dilations

Example 1: Given triangle ABC with A (–2,0), B (0, 4) and C (2, 1) increase the size of the triangle by a factor of 1.5. Then, sketch the image. 1) Dilations Vertices of the Preimage A B C

Example 1: Given triangle ABC with A (–2,0), B (0, 4) and C (2, 1) increase the size of the triangle by a factor of 1.5. Then, sketch the image. 1) Dilations Scalar multiplication Vertices of the Preimage A B C

Example 1: Given triangle ABC with A (–2,0), B (0, 4) and C (2, 1) increase the size of the triangle by a factor of 1.5. Then, sketch the image. 1) Dilations Vertices of the Preimage A B C Vertices of the Image A’ B’ C’ Scalar multiplication

1) Dilations

Example 2: Reduce the figure by a factor of ½. Find the coordinates of the image. 1) Dilations

Vertices of the Preimage A B C D

1) Dilations Vertices of the Preimage A B C D Vertices of the Image A’ B’ C’ D’

1) Dilations Vertices of the Preimage A B C D Vertices of the Image A’ B’ C’ D’

Homework p.195 #1, 2, 5, 6, 9, 26, 27-30, 35, 39, 40 Homework due at end of class tomorrow. Quiz tomorrow – Section 4.3 Matrix Multiplication Quiz Prep p.189 #57-59

Quiz Prep Answers p.189 # ) B 58) H 59)