10.15: 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.
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: \begin{align*}T_{a,b}:(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: \begin{align*}r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*}.
- Rotation: \begin{align*}R_{90^\circ}(x,y)=(-y,x)\end{align*}
Example A
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 satisfies the rule \begin{align*}r_{y-axis} \ \circ \ R_{90^\circ}\end{align*}.
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*}.
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 with mapping rule \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 \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 with mapping rule \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 \begin{align*}y\end{align*}-axis. The notation for this is \begin{align*}r_{y-axis}\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. The notation for this transformation is \begin{align*}R_{270^\circ}\end{align*}. Therefore, the notation to describe the transformation of Image A to Image C is \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 \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 satisfies the rule \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 \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*}.
Answers:
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*}. Notation for this composite transformation is:
\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 \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*}. Notation for this composite transformation is:
\begin{align*}r_{y-axis} \ \circ \ T_{-1,-5}\end{align*}
Practice
Complete the following table:
Starting Point | \begin{align*}T_{3,-4} \ \circ \ R_{90^{\circ}}\end{align*} | \begin{align*}r_{x-axis} \ \circ \ r_{y-axis}\end{align*} | \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.
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.Image Attributions
Here you will learn notation for describing a composite transformation.
Concept Nodes:
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.