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10.15: Notation for Composite Transformations

Difficulty Level: At Grade Created by: CK-12
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The figure below shows a composite transformation of a trapezoid. Write the mapping rule for the composite transformation.

Watch This

First watch this video to learn about notation for composite transformations.

CK-12 Foundation Chapter10NotationforCompositeTransformationsA

Then watch this video to see some examples.

CK-12 Foundation Chapter10NotationforCompositeTransformationsB

Guidance

In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A composite transformation is when two or more transformations are performed on a figure (called the preimage) to produce a new figure (called the image). The order of transformations performed in a composite transformation matters.

To describe a composite transformation using notation, state each of the transformations that make up the composite transformation and link them with the symbol . The transformations are performed in order from right to left. Recall the following notation for translations, reflections, and rotations:

  • Translation: is a translation of units to the right and units up.
  • Reflection: .
  • Rotation:

Example A

Graph the line given that and . Also graph the composite image that satisfies the rule .

Solution: The first translation is a CCW turn about the origin to produce . The second translation is a reflection about the -axis to produce .

Example B

Image A with vertices and undergoes a composite transformation with mapping rule . Draw the preimage and the composite image and show the vertices of the composite image.

Solution:

Example C

Image D with vertices and undergoes a composite transformation with mapping rule . Draw the preimage and the composite image and show the vertices of the composite image.

Solution:

Concept Problem Revisited

The transformation from Image A to Image B is a reflection across the -axis. The notation for this is . The transformation for image B to form image C is a rotation about the origin of CW. The notation for this transformation is . Therefore, the notation to describe the transformation of Image A to Image C is .

Vocabulary

Image
In a transformation, the final figure is called the image.
Preimage
In a transformation, the original figure is called the preimage.
Transformation
A transformation is an operation that is performed on a shape that moves or changes it in some way. There are four types of transformations: translations, reflections, dilations and rotations.
Dilation
A dilation is a transformation that enlarges or reduces the size of a figure.
Translation
A translation is an example of a transformation that moves each point of a shape the same distance and in the same direction. Translations are also known as slides.
Rotation
A rotation is a transformation that rotates (turns) an image a certain amount about a certain point.
Reflection
A reflection is an example of a transformation that flips each point of a shape over the same line.
Composite Transformation
A composite transformation is when two or more transformations are combined to form a new image from the preimage.

Guided Practice

1. Graph the line given that and . Also graph the composite image that satisfies the rule .

2. Describe the composite transformations in the diagram below and write the notation to represent the transformation of figure to .

3. Describe the composite transformations in the diagram below and write the notation to represent the transformation of figure to .

Answers:

1. The first transformation is a reflection about the -axis to produce . The second transformation is a CCW turn about the origin to produce .

2. There are two transformations shown in the diagram. The first transformation is a reflection about the line to produce . The second transformation is a CW (or CCW) rotation about the point (2, 0) to produce the figure . Notation for this composite transformation is:

3. There are two transformations shown in the diagram. The first transformation is a translation of 1 unit to the left and 5 units down to produce . The second reflection in the -axis to produce the figure . Notation for this composite transformation is:

Practice

Complete the following table:

Starting Point
1. (1, 4)
2. (4, 2)
3. (2, 0)
4. (-1, 2)
5. (-2, -3)
6. (4, -1)
7. (3, -2)
8. (5, 4)
9. (-3, 7)
10. (0, 0)

Write the notation that represents the composite transformation of the preimage A to the composite images in the diagrams below.

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Vocabulary

TermDefinition
Reflections Reflections are transformations that result in a "mirror image" of a parent function. They are caused by differing signs between parent and child functions.
Rotation A rotation is a transformation that turns a figure on the coordinate plane a certain number of degrees about a given point without changing the shape or size of the figure.
Transformation A transformation moves a figure in some way on the coordinate plane.

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Difficulty Level:
At Grade
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Date Created:
Apr 30, 2013
Last Modified:
Feb 28, 2017
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MAT.GEO.958.L.1