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10.15: Notation for Composite Transformations

Difficulty Level: Advanced 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: Ta,b:(x,y)(x+a,y+b) is a translation of a units to the right and b units up.
  • Reflection: ryaxis(x,y)(x,y).
  • Rotation: R90(x,y)=(y,x)

Example A

Graph the line XY given that X(2,2) and Y(3,4). Also graph the composite image that satisfies the rule

ryaxis  R90
.

Solution: The first translation is a 90CCW turn about the origin to produce XY. The second translation is a reflection about the y-axis to produce XY.

Example B

Image A with vertices A(3,5),B(4,2) and C(1,1) undergoes a composite transformation with mapping rule rxaxis  ryaxis. Draw the preimage and the composite image and show the vertices of the composite image.

Solution:

Example C

Image D with vertices D(3,7),E(1,3),F(7,5) and G(5,1) undergoes a composite transformation with mapping rule T3,4  rxaxis. 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 y-axis. The notation for this is ryaxis. The transformation for image B to form image C is a rotation about the origin of 90CW. The notation for this transformation is R270. Therefore, the notation to describe the transformation of Image A to Image C is

R270  ryaxis
.

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 XY given that X(2,2) and Y(3,4). Also graph the composite image that satisfies the rule

R90  ryaxis
.

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

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

Answers:

1. The first transformation is a reflection about the y-axis to produce XY. The second transformation is a 90CCW turn about the origin to produce XY.

2. There are two transformations shown in the diagram. The first transformation is a reflection about the line X=2 to produce ABCD. The second transformation is a 90CW (or 270CCW) rotation about the point (2, 0) to produce the figure ABCD. Notation for this composite transformation is:

R270  rx=2

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 ABC. The second reflection in the y-axis to produce the figure ABC. Notation for this composite transformation is:

ryaxis  T1,5

Practice

Complete the following table:

Starting Point T3,4  R90 rxaxis  ryaxis T1,6  rxaxis \begin{align*}r_{y-axis} \ \circ \ R_{180^{\circ}}\end{align*}
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.

Vocabulary

Reflections

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

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

Transformation

A transformation moves a figure in some way on the coordinate plane.

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Difficulty Level:

Advanced

Grades:

Date Created:

Apr 30, 2013

Last Modified:

Sep 21, 2015
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