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
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
A B Cx
y
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
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
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”
1) Translations
Image
Preimage
1) Translations
Example 2:
Translate the image 2 units left and 1 unit up
40167
25238
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’
1) Translations and Dilations with Matrices
Example 2:
40167
25238
31278
07416
Preimage
Image
1) Translations and Dilations with Matrices
Example 2:
40167
25238
31278
07416
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
1) Dilations
1) Dilations
2565
0505
Vertices of the Preimage
A B C D
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’
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
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
Quiz Prep Answers
p.189 #57-59
57) B
58) H
59)
20
15Y
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