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# Order of Composite Transformations

## Understand the order in which to apply transformations

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Order of Composite Transformations

Quadrilateral WXYZ\begin{align*}WXYZ\end{align*} has coordinates W(5,5),X(2,0),Y(2,3)\begin{align*}W(-5, -5), X(-2, 0), Y(2, 3)\end{align*} and Z(1,3)\begin{align*}Z(-1, 3)\end{align*}. Draw the quadrilateral on the Cartesian plane.

The quadrilateral undergoes a dilation centered at the origin of scale factor 13\begin{align*}\frac{1}{3}\end{align*} and then is translated 4 units to the right and 5 units down. Show the resulting image.

### Watch This

First watch this video to learn about the order of composite transformations.

CK-12 Foundation Chapter10OrderofCompositeTransformationsA

Then watch this video to see some examples.

CK-12 Foundation Chapter10OrderofCompositeTransformationsB

### 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).

Imagine if you rotate, then dilate, and then translate a rectangle of vertices A(1,1),B(1,3),C(5,3)\begin{align*}A(1, 1), B(1, 3), C(5, 3)\end{align*}, and D(5,1)\begin{align*}D(5, 1)\end{align*}. You would end up with a diagram similar to that found below:

If you take the same preimage and rotate, translate it, and finally dilate it, you could end up with the following diagram:

Therefore, the order is important when performing a composite transformation. Remember that the composite transformation involves a series of one or more transformations in which each transformation after the first is performed on the image that was transformed. Only the first transformation will be performed on the initial preimage.

#### Example A

Line AB¯¯¯¯¯\begin{align*}\overline{AB}\end{align*} drawn from (-4, 2) to (3, 2) has undergone a reflection across the x\begin{align*}x\end{align*}-axis. It then undergoes a translation up one unit and over 3 units to the right to produce \begin{align*}A^{\prime \prime} B^{\prime \prime}\end{align*}. Draw a diagram to represent this composite transformation and indicate the vertices for each transformation.

Solution:

#### Example B

For the composite transformation in Example A, suppose the preimage \begin{align*}AB\end{align*} undergoes a translation up one unit and over 3 units to the right and then undergoes a reflection across the \begin{align*}x\end{align*}-axis. Does the order matter?

Solution:

For this example \begin{align*}A^{\prime \prime} B^{\prime \prime}\end{align*} is not the same as \begin{align*}A^{\prime \prime} B^{\prime \prime}\end{align*} from the previous example (example A). Therefore order does matter.

#### Example C

Triangle \begin{align*}BCD\end{align*} is rotated \begin{align*}90^\circ\end{align*}CCW about the origin. The resulting figure is then translated over 3 to the right and down 7. Does order matter?

Order: Rotation then Translation

Order: Translation then Rotation

Solution: The blue triangle represents the final image after the composite transformation. In this example, order does matter as the blue triangles do not end up in the same locations.

#### Concept Problem Revisited

Quadrilateral \begin{align*}WXYZ\end{align*} has coordinates \begin{align*}W(-5, -5), X(-2, 0), Y(2, 3)\end{align*} and \begin{align*}Z(-1, 3)\end{align*}. Draw the quadrilateral on the Cartesian plane.

The quadrilateral undergoes a dilation centered at the origin of scale factor \begin{align*}\frac{1}{3}\end{align*} and then is translated 4 units to the right and 5 units down. Show the resulting image.

### 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. Line \begin{align*}\overline{ST}\end{align*} drawn from (-3, 4) to (-3, 8) has undergone a rotation about the origin at \begin{align*}90^\circ \end{align*}CW and then a reflection in the \begin{align*}x\end{align*}-axis. Draw a diagram with labeled vertices to represent this composite transformation.

2. Line \begin{align*}\overline{ST}\end{align*} drawn from (-3, 4) to (-3, 8) has undergone a reflection in the \begin{align*}x\end{align*}-axis and then a rotation about the origin at \begin{align*}90^\circ\end{align*}CW. Draw a diagram with labeled vertices to represent this composite transformation. Is the graph the same as the diagram in #1?

3. The triangle with vertices \begin{align*}J(-5, -2), K (-1, 4)\end{align*} and \begin{align*}L (1, -3)\end{align*} has undergone a transformation of up 4 and over to the right 4 and then a reflection in the \begin{align*}x\end{align*}-axis. Draw and label the composite transformation. Does order matter?

1.

2.

If you compare the graph above to that found in Question 1, you see that the final transformation image \begin{align*}S^{\prime \prime}T^{\prime \prime}\end{align*} has different coordinates than the image \begin{align*}S^{\prime \prime}T^{\prime \prime}\end{align*} in question 2. Therefore order does matter.

3.

Order: Translation then Reflection

Order: Reflection then Transformation

In this problem, order did matter. The final image after the composite transformation changed when the order changed.

### Practice

1. Reflect the above figure across the x-axis and then rotate it \begin{align*}90^\circ\end{align*} CW about the origin.
2. Translate the above figure 2 units to the left and 2 units up and then reflect it across the line \begin{align*}y=x\end{align*}

1. Reflect the above figure across the y-axis and then reflect it across the x-axis.
2. What single transformation would have produced the same result as in #3?

1. Reflect the above figure across the line \begin{align*}x=2\end{align*} and then across the line \begin{align*} x=8\end{align*}.
2. What single transformation would have produced the same result a in #5?

1. Reflect the above figure across the line y=x and then across the x-axis.
2. What single transformation would have produced the same result a in #7?

1. Translate the above figure 2 units to the right and 3 units down and then reflect it across the y-axis.
2. Rotate the above figure \begin{align*}270^\circ\end{align*} CCW about the origin and then translate it over 1 unit to the right and down 1 unit.

1. Reflect the above figure across the line \begin{align*}y=-x\end{align*} and then translate it 2 units to the left and 3 units down.
2. Translate the above figure 2 units to the left and 3 units down and then reflect it across the line \begin{align*}y=-x\end{align*}.

1. Translate the above figure 3 units to the right and 4 units down and then rotate it about the origin \begin{align*}90^\circ\end{align*} CW.
2. Rotate the above figure about the origin \begin{align*}90^\circ\end{align*} CW and then translate it 3 units to the right and 4 units down.
3. How did your result to #13 compare to your result to #14?

### Vocabulary Language: English

Dilation

Dilation

To reduce or enlarge a figure according to a scale factor is a dilation.