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# Notation for Composite Transformations

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Notation for Composite Transformations

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.

Then watch this video to see some examples.

### 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 $A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}$ .

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.

### Vocabulary Language: English

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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