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.
CK12 Foundation Chapter10NotationforCompositeTransformationsA
Then watch this video to see some examples.
CK12 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
 Translation:
Ta,b:(x,y)→(x+a,y+b) is a translation ofa units to the right andb units up.  Reflection:
ry−axis(x,y)→(−x,y) .  Rotation:
R90∘(x,y)=(−y,x)
Example A
Graph the line
Solution: The first translation is a
Example B
Image A with vertices
Solution:
Example C
Image D with vertices
Solution:
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.
 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
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
Complete the following table:
Starting Point 



\begin{align*}r_{yaxis} \ \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
Description
Learning Objectives
Here you will learn notation for describing a composite transformation.