4.4 transformations with matrices 1.translations and dilations 2.reflections and rotations

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

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

4.4 Transformations with Matrices

1. Translations and Dilations

2. Reflections and Rotations

Page 2: 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. 

Page 3: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

1) Translations and Dilations with Matrices

Translation

Preimage

Image

Moved the preimage…

Size and shape did not change

Page 4: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

1) Translations and Dilations with Matrices

Dilation

Preimage Image

Enlarged the preimage…

Size and shape did change

Page 5: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

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

Page 6: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

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

Page 7: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

Step 1: Write the vertices of the triangle as a matrix

1) Translations

Vertices of the Triangle

A B Cx

y

Page 8: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

Step 2: Write a translation matrix

1) Translations

Vertices of the Triangle

A B C

Add 5 to each x-coordinate

Subtract 3 from each y-coordinate

x

y

Page 9: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

Step 3: Add the matrices

1) Translations

Vertices of the Triangle

A B C

Add 5 to each x-coordinate

Subtract 3 from each y-coordinate

Vertices of the Image

A’ B’ C’x

y

Page 10: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

Step 3: Add the matrices

1) Translations

Vertices of the Triangle

A B C

Add 5 to each x-coordinate

Subtract 3 from each y-coordinate

Vertices of the Image

A’ B’ C’x

y

“A prime”

“B prime”

“C prime”

Page 11: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

                                                                                             

                                                                                                   

                                                                                             

                                                                                                   

                                                                                                   

1) Translations

Image

Preimage

Page 12: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

1) Translations

Example 2:

Translate the image 2 units left and 1 unit up

40167

25238

Page 13: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

1) Translations

Example 2:

11111

22222

Vertices of the Preimage

A B C D ESubtract 2 from

each x-coordinate

Add 1 to each y-coordinate

40167

25238+ =

31278

07416

Vertices of the Image

A’ B’ C’ D’ E’

Page 14: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

1) Translations and Dilations with Matrices

Example 2:

40167

25238

31278

07416

Preimage

Image

Page 15: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

1) Translations and Dilations with Matrices

Example 2:

40167

25238

31278

07416

Preimage

Image

Page 16: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

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

Page 17: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

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

Page 18: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

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

Page 19: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

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

Page 20: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

1) Dilations

Page 21: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

Example 2:

Reduce the figure by a factor of ½. Find the coordinates of the image.

1) Dilations

Page 22: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

1) Dilations

Page 23: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

1) Dilations

2565

0505

Vertices of the Preimage

A B C D

Page 24: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

1) Dilations

15.235.2

05.205.2

2565

05055.0

Vertices of the Preimage

A B C D

Vertices of the Image

A’ B’ C’ D’

Page 25: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

1) Dilations

Vertices of the Preimage

A B C D

Vertices of the Image

A’ B’ C’ D’

15.235.2

05.205.2

2565

05055.0

Page 26: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

Homework

p.195 #1, 2, 5, 6, 9, 26, 27-30, 35, 39, 40

Homework due at end of class tomorrow.

Quiz tomorrow – Section 4.3Matrix MultiplicationQuiz Prepp.189 #57-59

Page 27: 4.4 Transformations with Matrices 1.Translations and Dilations 2.Reflections and Rotations

Quiz Prep Answers

p.189 #57-59

57) B

58) H

59)

20

15Y