Compositions of Transformations

John Williams, Sep 19, 2016

Theorems and Investigations regarding compositions of transformations

Table of Contents

  1. Properties ofTransformations
    1. Chapter Overview
    2. Translations
    3. Reflections
    4. Rotations
    5. Chapter Closure
  2. Compositions of Two of the Same Transformation
    1. A Translation and a Translation
    2. A Reflection and a Reflection
    3. A Rotation and a Rotation
  3. Compositions of Two Different Transformations
    1. A Translation and a Reflection
    2. A Translation and a Rotation
    3. A Reflection and a Rotation

1.

Properties ofTransformations

  • 1. Chapter Overview
  • 2. Translations
  • 3. Reflections
  • 4. Rotations
  • 5. Chapter Closure

1.1.

Chapter Overview

Goals
Before we can test the conjectures we made about the compositions of transformations we did using patty paper, we need to have a surefire method of knowing whether or not a single transformation has occurred or not. This chapter will walk you through a test for each of the transformations. First, I'd like you to summarize what methods you already know about.
Translations
How can you tell that a transformation might be a translation? How can you tell a transformation is not a translation?
Reflections
How can you tell that a transformation might be a reflection? How can you tell a transformation is not a reflection?
Rotations
How can you tell that a transformation might be a rotation? How can you tell a transformation is not a rotation?
– John Williams

1.2.

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1.3.

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1.4.

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1.5.

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2.

Compositions of Two of the Same Transformation

  • 1. A Translation and a Translation
  • 2. A Reflection and a Reflection
  • 3. A Rotation and a Rotation

2.1.

A Translation and a Translation

Explore
A quadrilateral has been translated by first one vector, and then another. Drag the vectors to explore the transformation. Use the tests explained in the previous chapter to find out whether or not the composition could be reduced to a single transformation.
Conjecture
Restate your conjecture regarding translations from the patty paper sheet.
– John Williams

2.2.

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2.3.

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3.

Compositions of Two Different Transformations

  • 1. A Translation and a Reflection
  • 2. A Translation and a Rotation
  • 3. A Reflection and a Rotation

3.1.

A Translation and a Reflection

Explore
A quadrilateral has been translated by a vector and reflected over line. Move the points that control the vector and the line around to explore the composition. Use the tests of transformations to decide whether or not the composition could be formed by a single transformation.
Conjecture
Restate your conjecture from the patty paper worksheet regarding the composition of a translation and a reflection.
– John Williams

3.2.

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3.3.

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