<meta http-equiv="refresh" content="1; url=/nojavascript/"> Notation for Composite Transformations | CK-12 Foundation
Dismiss
Skip Navigation
You are reading an older version of this FlexBook® textbook: CK-12 Algebra I Concepts - Honors Go to the latest version.

10.15: Notation for Composite Transformations

Created by: CK-12
 0  0  0

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 \circ . The transformations are performed in order from right to left. Recall the following notation for translations, reflections, and rotations:

  • Translation: T_{a,b}:(x, y) \rightarrow  (x + a, y + b) is a translation of a units to the right and b units up.
  • Reflection: r_{y-axis}(x,y) \rightarrow (-x,y) .
  • Rotation: R_{90^\circ}(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 r_{y-axis} \ \circ \ R_{90^\circ} .

Solution: The first translation is a 90^{\circ} CCW turn about the origin to produce X^{\prime}Y^{\prime} . The second translation is a reflection about the y -axis to produce X^{\prime \prime}Y^{\prime \prime} .

Example B

Image A with vertices A(3, 5), B(4, 2) and C(1, 1) undergoes a composite transformation with mapping rule r_{x-axis} \ \circ \ r_{y-axis} . 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 T_{3,4} \ \circ \ r_{x-axis} . 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 r_{y-axis} . The transformation for image B to form image C is a rotation about the origin of 90^\circ CW. The notation for this transformation is R_{270^\circ} . Therefore, the notation to describe the transformation of Image A to Image C is R_{270^\circ}\ \circ \ r_{y-axis} .

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 R_{90^\circ} \ \circ \ r_{y-axis} .

2. Describe the composite transformations in the diagram below and write the notation to represent the transformation of figure ABCD to A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}D^{\prime \prime} .

3. Describe the composite transformations in the diagram below and write the notation to represent the transformation of figure ABC to A^{\prime \prime}B^{\prime \prime}C^{\prime \prime} .

Answers:

1. The first transformation is a reflection about the y -axis to produce X^{\prime}Y^{\prime} . The second transformation is a 90^\circ CCW turn about the origin to produce X^{\prime \prime}Y^{\prime \prime} .

2. There are two transformations shown in the diagram. The first transformation is a reflection about the line X = 2 to produce A^{\prime}B^{\prime}C^{\prime}D^{\prime} . The second transformation is a 90^{\circ} CW (or 270^{\circ} CCW) rotation about the point (2, 0) to produce the figure A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}D^{\prime \prime} . Notation for this composite transformation is:

R_{270^{\circ}} \ \circ \ r_{x=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 A^{\prime}B^{\prime}C^{\prime} . The second reflection in the y -axis to produce the figure A^{\prime \prime}B^{\prime \prime}C^{\prime \prime} . Notation for this composite transformation is:

r_{y-axis} \ \circ \ T_{-1,-5}

Practice

Complete the following table:

Starting Point T_{3,-4} \ \circ \ R_{90^{\circ}} r_{x-axis} \ \circ \ r_{y-axis} T_{1,6} \ \circ \ r_{x-axis} r_{y-axis} \ \circ \ R_{180^{\circ}}
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.

Image Attributions

Description

Difficulty Level:

Advanced

Grades:

Date Created:

Apr 30, 2013

Last Modified:

Jul 15, 2014

We need you!

At the moment, we do not have exercises for Notation for Composite Transformations.

Files can only be attached to the latest version of Modality

Reviews

Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.GEO.958.L.1
ShareThis Copy and Paste

Original text