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 . 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 about the y-axis: .
- 90° Rotation:
Example A
Graph the line segment given that and . Reflect the line segment about the y-axis and then rotate it 90°.
Solution: The first translation is a CCW turn about the origin to produce . The second translation is a reflection about the -axis to produce . The transformation from preimage to image is .
Example B
Image A with vertices and 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 .
Example C
Image D with vertices and 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 .
Concept Problem Revisited
The transformation from Image A to Image B is a reflection across the -axis or . The transformation for image B to form image C is a rotation about the origin of CW or . 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.
- 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 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 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 .
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 .
Practice
Write the notation that represents the composite transformation of the preimage A to the composite images in the diagrams below.