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
- Translation:
(x,y)→(x+a,y+b) is a translation ofa units to the right andb units up. - Reflection about the y-axis:
(x,y)→(−x,y) . - 90° Rotation:
(x,y)=(−y,x)
Example A
Graph the line segment
Solution: The first translation is a
Example B
Image A with vertices
Solution:
The transformation from preimage to image is
Example C
Image D with vertices
Solution:
The transformation from preimage to image is
Concept Problem Revisited
The transformation from Image A to Image B is a reflection across the
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
2. Describe the composite transformations in the diagram below and write the notation to represent the transformation of figure
3. Describe the composite transformations in the diagram below and write the notation to represent the transformation of figure
Answers:
1. The first transformation is a reflection about the
2. There are two transformations shown in the diagram. The first transformation is a reflection about the line
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
Practice
Write the notation that represents the composite transformation of the preimage A to the composite images in the diagrams below.