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Composition of Transformations

Learn how to compose transformations of a figure on a coordinate plane.

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Composite Transformations

Look at the following diagram. It involves two translations. Identify the two translations of triangle .

Watch This

First watch this video to learn about composite transformations.

CK-12 Foundation Chapter10CompositeTransformationsA

Then watch this video to see some examples.

CK-12 Foundation Chapter10CompositeTransformationsB

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).

Example A

Describe the transformations in the diagram below. The transformations involve a reflection and a rotation.

Solution:First line is rotated about the origin by CCW.

Then the line is reflected about the -axis to produce line .

Example B

Describe the transformations in the diagram below.

Solution: The flag in diagram S is rotated about the origin to produce flag T. You know this because if you look at one point you notice that both - and -coordinate points is multiplied by -1 which is consistent with a rotation about the origin. Flag T is then reflected about the line to produce Flag U.

Example C

Triangle where the vertices of are , , and undergoes a composition of transformations described as:

a) a translation 10 units to the right, then

b) a reflection in the -axis.

Draw the diagram to represent this composition of transformations. What are the vertices of the triangle after both transformations are applied?

Solution:

Triangle is the final triangle after all transformations are applied. It has vertices of , , and .

Concept Problem Revisited

moves over 6 to the left and down 5 to produce . Then moves over 14 to the right and up 3 to produce . These translations are represented by the blue arrows in the diagram.

All together moves over 8 to the right and down 2 to produce . The total translations for this movement are seen by the green arrow in the diagram above.

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. Describe the transformations in the diagram below. The transformations involve a rotation and a reflection.

2. Triangle has coordinates , and .The triangle undergoes a rotation of 2 units to the right and 1 unit down to form triangle . Triangle is then reflected about the -axis to form triangle . Draw the diagram of this composite transformation and determine the vertices for triangle .

3. The coordinates of the vertices of are , , and .

a) Draw and label .

b) is reflected over the line . Graph and state the coordinates of .

c) is then reflected about the -axis. Graph and state the coordinates of .

d) undergoes a translation of 5 units to the left and 3 units up. Graph and state the coordinates of .

1. The transformations involve a reflection and a rotation. First line is reflected about the -axis to produce line .

Then the line is rotated about the origin by CCW to produce line .

2.

3.

Practice

1. A point has coordinates (-1, -8). The point is reflected across the -axis to form . is translated over 4 to the right and up 6 to form . What are the coordinates of and ?
2. A point has coordinates (2, -3). The point is translated over 3 to the left and up 5 to form . is reflected across the -axis to form . What are the coordinates of and ?
3. A point has coordinates (5, -6). The point is reflected across the line to form . is rotated about the origin CW to form . What are the coordinates of and ?
4. Line has coordinates and . The segment is rotated about the origin to form . is translated over 6 to the right and down 3 to form . What are the coordinates of and ?
5. Line has coordinates and . The segment is translated over 3 to the right and up 3 to form . is rotated about the origin CCW to form . What are the coordinates of and ?
6. A point has coordinates (-1, 4). The point is reflected across the line to form . is rotated about the origin CW to form . What are the coordinates of and ?

Describe the following composite transformations:

1. Explore what happens when you reflect a shape twice, over a pair of parallel lines. What one transformation could have been performed to achieve the same result?
2. Explore what happens when you reflect a shape twice, over a pair of intersecting lines. What one transformation could have been performed to achieve the same result?
3. Explore what happens when you reflect a shape over the x-axis and then the y-axis. What one transformation could have been performed to achieve the same result?
4. A composition of a reflection and a translation is often called a glide reflection. Make up an example of a glide reflection. Why do you think it's called a glide reflection?

Vocabulary Language: English

Reflection

Reflection

A reflection is a transformation that flips a figure on the coordinate plane across a given line without changing the shape or size of the figure.
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.
Glide Reflection

Glide Reflection

A reflection followed by a translation where the line of reflection is parallel to the direction of translation is called a glide reflection or a walk.
Composite Transformation

Composite Transformation

A composite transformation, also known as composition of transformation, is a series of multiple transformations performed one after the other.