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# Notation for Composite Transformations

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Combining Transformations

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

• Translation: $(x, y) \rightarrow (x + a, y + b)$ is a translation of $a$ units to the right and $b$ units up.
• Reflection about the y -axis: $(x,y) \rightarrow (-x,y)$ .
• 90° Rotation: $(x,y)=(-y,x)$

#### Example A

Graph the line segment  $XY$ given that $X(2, -2)$ and $Y(3, -4)$ . Reflect the line segment about the y-axis and then rotate it 90 °.

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 $X^{\prime \prime}Y^{\prime \prime}$ . The transformation from preimage to image is  $\mathit{}(x, y)\rightarrow(y, x)$ .

#### Example B

Image A with vertices $A(3, 5), B(4, 2)$ and $C(1, 1)$ 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  $\mathit{}(x, y)\rightarrow(-x, -y)$ .

#### Example C

Image D with vertices $D(-3, 7), E(-1, 3), F(-7, 5)$ and $G(-5, 1)$ 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  $\mathit{}(x, y)\rightarrow(x+3, -y+4)$ .

#### Concept Problem Revisited

The transformation from Image A to Image B is a reflection across the $y$ -axis or   $\mathit{}(x, y)\rightarrow(-x, y)$ . The transformation for image B to form image C is a rotation about the origin of $90^\circ$ CW or $\mathit{}(x, y)\rightarrow(y,-x)$ .  Therefore, the notation to describe the transformation of Image A to Image C is  $\mathit{}(x, y)\rightarrow(y,x)$ .

### 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 $XY$ given that $X(2, -2)$ and $Y(3, -4)$ . Also graph the composite image that shows a reflection about the y -axis and a counterclockwise rotation of 90 º .

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 $A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}D^{\prime \prime}$

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

### Practice

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