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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 \begin{align*}\circ\end{align*}. 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)\begin{align*}T_{a,b}:(x, y) \rightarrow (x + a, y + b)\end{align*} is a translation of a\begin{align*}a\end{align*} units to the right and b\begin{align*}b\end{align*} units up.
• Reflection: ryaxis(x,y)(x,y)\begin{align*}r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*}.
• Rotation: R90(x,y)=(y,x)\begin{align*}R_{90^\circ}(x,y)=(-y,x)\end{align*}

#### Example A

Graph the line XY\begin{align*}XY\end{align*} given that X(2,2)\begin{align*}X(2, -2)\end{align*} and Y(3,4)\begin{align*}Y(3, -4)\end{align*}. Also graph the composite image that satisfies the rule ryaxis  R90\begin{align*}r_{y-axis} \ \circ \ R_{90^\circ}\end{align*}.

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

#### Example B

Image A with vertices A(3,5),B(4,2)\begin{align*}A(3, 5), B(4, 2)\end{align*} and C(1,1)\begin{align*}C(1, 1)\end{align*} undergoes a composite transformation with mapping rule rxaxis  ryaxis\begin{align*}r_{x-axis} \ \circ \ r_{y-axis}\end{align*}. 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)\begin{align*}D(-3, 7), E(-1, 3), F(-7, 5)\end{align*} and G(5,1)\begin{align*}G(-5, 1)\end{align*} undergoes a composite transformation with mapping rule T3,4  rxaxis\begin{align*}T_{3,4} \ \circ \ r_{x-axis}\end{align*}. 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\begin{align*}y\end{align*}-axis. The notation for this is ryaxis\begin{align*}r_{y-axis}\end{align*}. The transformation for image B to form image C is a rotation about the origin of 90\begin{align*}90^\circ\end{align*}CW. The notation for this transformation is R270\begin{align*}R_{270^\circ}\end{align*}. Therefore, the notation to describe the transformation of Image A to Image C is R270  ryaxis\begin{align*}R_{270^\circ}\ \circ \ r_{y-axis}\end{align*}.

### 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\begin{align*}XY\end{align*} given that X(2,2)\begin{align*}X(2, -2)\end{align*} and Y(3,4)\begin{align*}Y(3, -4)\end{align*}. Also graph the composite image that satisfies the rule R90  ryaxis\begin{align*}R_{90^\circ} \ \circ \ r_{y-axis}\end{align*}.

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

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

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

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

R270  rx=2\begin{align*}R_{270^{\circ}} \ \circ \ r_{x=2}\end{align*}

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\begin{align*}A^{\prime}B^{\prime}C^{\prime}\end{align*}. The second reflection in the y\begin{align*}y\end{align*}-axis to produce the figure ABC\begin{align*}A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}\end{align*}. Notation for this composite transformation is:

ryaxis  T1,5\begin{align*}r_{y-axis} \ \circ \ T_{-1,-5}\end{align*}

### Practice

Complete the following table:

Starting Point T3,4  R90\begin{align*}T_{3,-4} \ \circ \ R_{90^{\circ}}\end{align*} rxaxis  ryaxis\begin{align*}r_{x-axis} \ \circ \ r_{y-axis}\end{align*} T1,6  rxaxis\begin{align*}T_{1,6} \ \circ \ r_{x-axis}\end{align*} \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.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

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

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

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

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