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# Notation for Composite Transformations

## Interpret and use notation for combined transformations

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Practice Notation for Composite Transformations
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SLT 17 Combine transformations and write the associated function.

The figure below shows a composite transformation of a trapezoid. Write the mapping rule for the composite transformation.

### 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 about the y-axis: .
• 90° Rotation:

#### Example A

Graph the line segment  given that and . Reflect the line segment about the y-axis and then rotate it 90°.

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

#### Example B

Image A with vertices and undergoes a composite transformation that is a reflection about the x-axis and then a reflection about the y-axis. Draw the preimage and the composite image and show the vertices of the composite image.

Solution:

The transformation from preimage to image is .

#### Example C

Image D with vertices and undergoes a composite transformation that is a horizontal translation to the right 3, a vertical translation up 4, and a reflection about the x-axis. Draw the preimage and the composite image and show the vertices of the composite image.

Solution:

The transformation from preimage to image is .

#### Concept Problem Revisited

The transformation from Image A to Image B is a reflection across the -axis or . The transformation for image B to form image C is a rotation about the origin of CW or .  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.
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 shows a reflection about the y-axis and a counterclockwise rotation of 90º.

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 .

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

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

### Practice

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