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 . The transformations are performed in order from right to left. Recall the following notation for translations, reflections, and rotations:
- Translation: is a translation of units to the right and units up.
- Reflection: .
- Rotation:
Example A
Graph the line given that and . Also graph the composite image that satisfies the rule .
Solution: The first translation is a CCW turn about the origin to produce . The second translation is a reflection about the -axis to produce .
Example B
Image A with vertices and undergoes a composite transformation with mapping rule . Draw the preimage and the composite image and show the vertices of the composite image.
Solution:
Example C
Image D with vertices and undergoes a composite transformation with mapping rule . 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 -axis. The notation for this is . The transformation for image B to form image C is a rotation about the origin of CW. The notation for this transformation is . Therefore, the notation to describe the transformation of Image A to Image C is .
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 given that and . Also graph the composite image that satisfies the rule .
2. Describe the composite transformations in the diagram below and write the notation to represent the transformation of figure to .
3. Describe the composite transformations in the diagram below and write the notation to represent the transformation of figure to .
Answers:
1. The first transformation is a reflection about the -axis to produce . The second transformation is a CCW turn about the origin to produce .
2. There are two transformations shown in the diagram. The first transformation is a reflection about the line to produce . The second transformation is a CW (or CCW) rotation about the point (2, 0) to produce the figure . Notation for this composite transformation is:
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 . The second reflection in the -axis to produce the figure . Notation for this composite transformation is:
Practice
Complete the following table:
Starting Point | ||||
---|---|---|---|---|
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.