Notation for Composite Transformations

Interpret and use notation for combined 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 \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: \begin{align*}(x, y) \rightarrow (x + a, y + b)\end{align*} is a translation of \begin{align*}a\end{align*} units to the right and \begin{align*}b\end{align*} units up.
• Reflection about the y-axis: \begin{align*}(x,y) \rightarrow (-x,y)\end{align*}.
• 90° Rotation: \begin{align*}(x,y)=(-y,x)\end{align*}

Example A

Graph the line segment \begin{align*}XY\end{align*} given that \begin{align*}X(2, -2)\end{align*} and \begin{align*}Y(3, -4)\end{align*}. Reflect the line segment about the y-axis and then rotate it 90°.

Solution: The first translation is a \begin{align*}90^{\circ}\end{align*}CCW turn about the origin to produce \begin{align*}X^{\prime}Y^{\prime}\end{align*}. The second translation is a reflection about the \begin{align*}y\end{align*}-axis to produce \begin{align*}X^{\prime \prime}Y^{\prime \prime}\end{align*}. The transformation from preimage to image is \begin{align*}\mathit{}(x, y)\rightarrow(y, x)\end{align*}.

Example B

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

Example C

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

Concept Problem Revisited

The transformation from Image A to Image B is a reflection across the \begin{align*}y\end{align*}-axis or \begin{align*}\mathit{}(x, y)\rightarrow(-x, y)\end{align*}. The transformation for image B to form image C is a rotation about the origin of \begin{align*}90^\circ\end{align*}CW or \begin{align*}\mathit{}(x, y)\rightarrow(y,-x)\end{align*}.  Therefore, the notation to describe the transformation of Image A to Image C is \begin{align*}\mathit{}(x, y)\rightarrow(y,x)\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.
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 \begin{align*}XY\end{align*} given that \begin{align*}X(2, -2)\end{align*} and \begin{align*}Y(3, -4)\end{align*}. 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 \begin{align*}ABCD\end{align*} to \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 \begin{align*}ABC\end{align*} to \begin{align*}A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}\end{align*}.

1. The first transformation is a reflection about the \begin{align*}y\end{align*}-axis to produce \begin{align*}X^{\prime}Y^{\prime}\end{align*}. The second transformation is a \begin{align*}90^\circ\end{align*}CCW turn about the origin to produce \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 \begin{align*}X = 2\end{align*} to produce \begin{align*}A^{\prime}B^{\prime}C^{\prime}D^{\prime}\end{align*}. The second transformation is a \begin{align*}90^{\circ}\end{align*}CW (or \begin{align*}270^{\circ}\end{align*}CCW) rotation about the point (2, 0) to produce the figure \begin{align*}A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}D^{\prime \prime}\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 \begin{align*}A^{\prime}B^{\prime}C^{\prime}\end{align*}. The second reflection in the \begin{align*}y\end{align*}-axis to produce the figure \begin{align*}A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}\end{align*}

Practice

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

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