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Notation for Composite Transformations

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Combining Transformations
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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 \circ . The transformations are performed in order from right to left. Recall the following notation for translations, reflections, and rotations:

  • Translation: (x, y) \rightarrow  (x + a, y + b) is a translation of a units to the right and b units up.
  • Reflection about the y -axis: (x,y) \rightarrow (-x,y) .
  • 90° Rotation: (x,y)=(-y,x)

Example A

Graph the line segment  XY given that X(2, -2) and Y(3, -4) . Reflect the line segment about the y-axis and then rotate it 90 °.

Solution: The first translation is a 90^{\circ} CCW turn about the origin to produce X^{\prime}Y^{\prime} . The second translation is a reflection about the y -axis to produce X^{\prime \prime}Y^{\prime \prime} . The transformation from preimage to image is  \mathit{}(x, y)\rightarrow(y, x) .


Example B

Image A with vertices A(3, 5), B(4, 2) and C(1, 1) 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  \mathit{}(x, y)\rightarrow(-x, -y) .


Example C

Image D with vertices D(-3, 7), E(-1, 3), F(-7, 5) and G(-5, 1) 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  \mathit{}(x, y)\rightarrow(x+3, -y+4) .


Concept Problem Revisited

The transformation from Image A to Image B is a reflection across the y -axis or   \mathit{}(x, y)\rightarrow(-x, y) . The transformation for image B to form image C is a rotation about the origin of 90^\circ CW or \mathit{}(x, y)\rightarrow(y,-x) .  Therefore, the notation to describe the transformation of Image A to Image C is  \mathit{}(x, y)\rightarrow(y,x) .

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 XY given that X(2, -2) and Y(3, -4) . 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 ABCD to A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}D^{\prime \prime} .


3. Describe the composite transformations in the diagram below and write the notation to represent the transformation of figure ABC to A^{\prime \prime}B^{\prime \prime}C^{\prime \prime} .

Answers:

1. The first transformation is a reflection about the y -axis to produce X^{\prime}Y^{\prime} . The second transformation is a 90^\circ CCW turn about the origin to produce X^{\prime \prime}Y^{\prime \prime} .

2. There are two transformations shown in the diagram. The first transformation is a reflection about the line X = 2 to produce A^{\prime}B^{\prime}C^{\prime}D^{\prime} . The second transformation is a 90^{\circ} CW (or 270^{\circ} CCW) rotation about the point (2, 0) to produce the figure A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}D^{\prime \prime}

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 A^{\prime}B^{\prime}C^{\prime} . The second reflection in the y -axis to produce the figure A^{\prime \prime}B^{\prime \prime}C^{\prime \prime}

Practice

Write the notation that represents the composite transformation of the preimage A to the composite images in the diagrams below.

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