4.4 Transformations with Matrices 2. Reflections and Rotations.

4.4 Transformations with Matrices

2. Reflections and Rotations

2) Reflections

A reflection, or flip, is a transformation that creates symmetry. 

You can use matrix multiplication to graph reflections in the coordinate plane.

There are four reflection matrices you are responsible for knowing.

2) Reflections

Reflection in the y-axis Reflection in the x-axis

10

01

10

01

2) Reflections

Reflection in the line y = x Reflection in the line y = -x

01

10

01

10

Example 1:  Given triangle ABC with A (–4, 1), B (– 2, 5) and C (0, 2), reflect the triangle across the y-axis. Then, sketch the image.

2) Reflections

A B C

Example 1:  Given triangle ABC with A (–4, 1), B (– 2, 5) and C (0, 2), reflect the triangle across the y-axis. Then, sketch the image.

2) Reflections

y-axis reflection matrix

A B C

Example 1:  Given triangle ABC with A (–4, 1), B (– 2, 5) and C (0, 2), reflect the triangle across the y-axis. Then, sketch the image.

2) Reflections

y-axis reflection matrix

A B C A’ B’ C’

2) Reflections

2) Reflections

Example 2: Given triangle ABC where A (–4, 1), B (– 2, 5) and C (0, 2), reflect the triangle across the x-axis. Then, sketch the image.

2) Reflections

Example 2: Given triangle ABC where A (–4, 1), B (– 2, 5) and C (0, 2), reflect the triangle across the x-axis. Then, sketch the image.

A B C

2) Reflections

Example 2: Given triangle ABC where A (–4, 1), B (– 2, 5) and C (0, 2), reflect the triangle across the x-axis. Then, sketch the image.

x-axis reflection matrix

A B C

2) Reflections

Example 2: Given triangle ABC where A (–4, 1), B (– 2, 5) and C (0, 2), reflect the triangle across the x-axis. Then, sketch the image.

251

024

251

024

10

01

x-axis reflection matrix

A B C A’ B’ C’

2) Reflections

251

024

251

024

10

01

A rotation is a transformation that turns a figure about a fixed point called a center of rotation. 

You can rotate a figure as much as 360o. 

In this text, all rotations are counterclockwise about the origin.

2) Rotations

2) Rotations

Rotation of 90o Rotation of 360o

Rotation of 180o Rotation of 270o

01

10

10

01

01

10

10

01

Example 1:  Given triangle ABC where A (–4, 1), B (– 2, 5) and C (0, 2), rotate the triangle 270°.  Then, sketch the image. 

2) Rotations

Example 1:  Given triangle ABC where A (–4, 1), B (– 2, 5) and C (0, 2), rotate the triangle 270°.  Then, sketch the image. 

2) Rotations

A B C

Example 1:  Given triangle ABC where A (–4, 1), B (– 2, 5) and C (0, 2), rotate the triangle 270°.  Then, sketch the image. 

2) Rotations

270o rotation matrtix

A B C

Example 1:  Given triangle ABC where A (–4, 1), B (– 2, 5) and C (0, 2), rotate the triangle 270°.  Then, sketch the image. 

2) Rotations

270o rotation matrtix

A B C A’ B’ C’

2) Rotations

2) Rotations

Example 2:  The matrix below represents the vertices of a polygon. Write a matrix to represent the vertices after a rotation of 90o.

A B C D

2) Rotations

Example 2:  The matrix below represents the vertices of a polygon. Write a matrix to represent the vertices after a rotation of 90o.

90o rotation matrtix

A B C D

2) Rotations

Example 2:  The matrix below represents the vertices of a polygon. Write a matrix to represent the vertices after a rotation of 90o.

90o rotation matrtix

A B C D A’ B’ C’ D’

Homework

1) Create some way to remember the 8 matrices used for reflections and rotations.

You are responsible for knowing all 8.

The matrices are located on p.193 and p.194

2) p.196 #10, 11, 13, 14, 18-21, 31, 32

3) QUIZ WEDNESDAY – section 4.4