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Introduction First we learned that transformations can be functions in the coordinate plane. Then we learned the definitions and properties of three isometric transformations: rotations, reflections, and translations. Now we are able to apply what we have learned to graph geometric figures and images created through transformations. 1 5.2.2: Applying Rotations, Reflections, and Translations

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Page 1: Introduction

IntroductionFirst we learned that transformations can be functions in the coordinate plane. Then we learned the definitions and properties of three isometric transformations: rotations, reflections, and translations. Now we are able to apply what we have learned to graph geometric figures and images created through transformations.

1

5.2.2: Applying Rotations, Reflections, and Translations

Page 2: Introduction

Key Concepts• Transformations can be precisely and accurately

graphed using the definitions learned.

• Given a set of points and a target, we can determine the transformation(s) necessary to move the given set of points to the target.

• Observing the orientations of the preimage and image is the first tool in determining the transformations required.

2

5.2.2: Applying Rotations, Reflections, and Translations

Page 3: Introduction

Key Concepts, continued• Graphs can be interpreted differently, allowing for many

transformation solution sets. While there are many different solution sets of transformations that will satisfy a particular graph, we will look for the more concise possibilities.

• Formulas can be used to determine translations, reflections, and rotations.

• Translation: Th, k(x, y) = (x + h, y + k)

3

5.2.2: Applying Rotations, Reflections, and Translations

Page 4: Introduction

Key Concepts, continued• Reflection:

• through the x-axis: rx-axis(x, y) = (x, –y)• through the y-axis: ry-axis(x, y) = (–x, y)• through the line y = x: ry = x(x, y) = (y, x)

• Rotation:• 90° rotation about the origin: R90(x, y) = (–y, x) • 180° rotation about the origin: R180(x, y) = (–x, –y) • 270° rotation about the origin: R270(x, y) = (y, –x)

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5.2.2: Applying Rotations, Reflections, and Translations

Page 5: Introduction

Common Errors/Misconceptions• using the incorrect transformation formula for

reflections

• using the incorrect transformation formula for rotations

• translating in the positive directions along the axes when the translations are intended to be in the negative directions

• applying the transformations in the wrong order

5

5.2.2: Applying Rotations, Reflections, and Translations

Page 6: Introduction

Guided PracticeExample 2Use the definitions you have learned to graph the reflection of parallelogram ABCD, or , through the y-axis given with the points A (–5, 5), B (–3, 4), C (–4, 1), and D (–6, 2).

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5.2.2: Applying Rotations, Reflections, and Translations

Page 7: Introduction

Guided Practice: Example 2, continued1. Using graph paper, draw the x- and y-axes

and graph with A (–5, 5), B (–3, 4), C (–4, 1), and D (–6, 2).

7

5.2.2: Applying Rotations, Reflections, and Translations

Page 8: Introduction

Guided Practice: Example 2, continued2. Write the new points.

where

8

5.2.2: Applying Rotations, Reflections, and Translations

Page 9: Introduction

Guided Practice: Example 2, continued3. Plot the new points , , , and .

9

5.2.2: Applying Rotations, Reflections, and Translations

Page 10: Introduction

Guided Practice: Example 2, continued4. Connect the corners of the points to

graph the reflection of .

10

5.2.2: Applying Rotations, Reflections, and Translations

✔

Page 11: Introduction

Guided Practice: Example 2, continued

11

5.2.2: Applying Rotations, Reflections, and Translations

http://walch.com/ei/CAU5L2S2Reflection
Page 12: Introduction

Guided PracticeExample 3Using the definitions you have learned, graph a 90° rotation of with the points A (1, 4), B (6, 3), and C (3, 1).

12

5.2.2: Applying Rotations, Reflections, and Translations

Page 13: Introduction

Guided Practice: Example 3, continued1. Using graph paper, draw the x- and y-axes

and graph with the points A (1, 4), B (6, 3), and C (3, 1).

13

5.2.2: Applying Rotations, Reflections, and Translations

Page 14: Introduction

Guided Practice: Example 3, continued2. Write the new points.

where

14

5.2.2: Applying Rotations, Reflections, and Translations

Page 15: Introduction

Guided Practice: Example 3, continued3. Plot the new points , , and .

15

5.2.2: Applying Rotations, Reflections, and Translations

Page 16: Introduction

Guided Practice: Example 3, continued4. Connect the vertices to graph a 90°

rotation of .

16

5.2.2: Applying Rotations, Reflections, and Translations

✔

Page 17: Introduction

Guided Practice: Example 3, continued

17

5.2.2: Applying Rotations, Reflections, and Translations

http://walch.com/ei/CAU5L2S2Rotation

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