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6. Rigid Motions
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Integrated Math 2

6.04 Compositions of transformations

Lesson
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Lesson

Concept summary

A composition of transformations is a list of transformations that are performed one after the other. These transformations include translations, rotations, and reflections.

When performing multiple transformations one after the other, the pre-image for each new transformation will be the image of the previous transformation.

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For example, suppose that we applied the following transformations to the point \left(-1,2\right):

  • Translate 4 units to the right
  • Rotate 90\degree clockwise about the origin
  • Reflect across the y-axis

We perform the translation first, then rotate the image of the translation, then reflect the image of the rotation.

When performing multiple transformations, the order in which they are applied matters.

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Suppose that we applied the following transformations to the point \left(-1,2\right):

  • Reflect across the y-axis
  • Rotate 90\degree clockwise about the origin
  • Translate 4 units to the right

As we can see, performing the transformations in a different order results in a different image of the transformations.

Worked examples

Example 1

Given the following transformations to the pre-image (5,-1), identify the coordinates of the image:

  • Rotated 90 \degree clockwise about the origin;
  • Translated 3 units up, and 2 units left;
  • Reflected across the y-axis.

Approach

We need to do the transformations in the order they are given.

  • Rotating the point 90 \degree clockwise is the same as rotating the point 270 \degree counterclockwise. The mapping for rotating something counterclockwise 270 \degree about the origin is \left(x,y \right) \to \left(y,-x\right).

  • Translating the point 3 units up and 2 units left has the mapping \left(x,y \right) \to \left(x-2,y+3\right).

  • Reflection across the y-axis has the mapping \left(x,y \right) \to \left(-x,y\right).

Solution

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  • First we rotate the point (5,-1) clockwise 90\degree about the origin to get the point (-1,-5).

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  • Next we translate the point (-1,-5) up three units and left two units to get (-3,-2).

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  • Finally we reflect the point (-3,-2) across the y-axis to get (3,-2).

Example 2

Describe the transformations required to obtain the image from the pre-image.

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Approach

First, we need to look to see if the side lengths are in the same order when looking at them clockwise, or in the reverse order. If the side lengths are in the same order when looking at them clockwise, then the pre-image is rotated. If they are in the reverse order, then the pre-image has been reflected.

Then, we need to see if it needs to be moved up, down, left or right.

Solution

The side lengths are in the reverse order, so a reflection is required. So, lets reflect the pre-image across the line y = -x.

Remember that when the line of reflection is y=-x, the transformation mapping is \left(x, y\right) \to \left(-y, -x \right)

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Now, the pre-image needs to be moved one unit to the left, and five units up.

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The transformations to the pre-image to get the image are:

  • Reflection across the line y = -x.
  • Translate 1 unit left, and 5 units up.

Reflection

There is more than one way to obtain an image from a pre-image using a composition of functions.

Another solution is:

  1. Reflect across the line x=3

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  2. Rotate 90\degree counterclockwise about the point \left(3,1\right).

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  3. Translate 5 units left and 1 unit up.

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Outcomes

M2.G.CO.A.1

Describe transformations as functions that take points in the plane (pre-image) as inputs and give other points (image) as outputs. Compare transformations that preserve distance and angle measure to those that do not, by hand for basic transformations and using technology for more complex cases.

M2.G.CO.A.4

Given a geometric figure, draw the image of the figure after a sequence of one or more rigid motions, by hand and using technology. Identify a sequence of rigid motions that will carry a given figure onto another.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP2

Reason abstractly and quantitatively.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP4

Model with mathematics.

M2.MP5

Use appropriate tools strategically.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

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