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8.19: Translations, Rotations, and Reflections

Difficulty Level: At Grade Created by: CK-12
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Practice Translations, Rotations, and Reflections
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Have you ever tried to create a piece of art? Look at what Tanya discovered at the art museum.

While at the art museum, Tanya found an area where you could create pieces of art using a computer. These graphics could be created and manipulated using a program. She used a computer to create this image.

Tanya has created an image that started in the blue position and moved to the red position.

Do you know what these two figures are called?

In the coordinate plane, you can change figures in many different ways. This Concept will teach you all about this.

Guidance

When we perform translations, we slide a figure left or right, up or down. This means that on the coordinate plane, the coordinates for the vertices of the figure will change. Take a look at the image below.

Now let’s look at performing a translation or slide of this figure.

We can choose the number places that we want to move the triangle and the direction that we wish to move it in. If we slide this triangle 3 places down, all of its vertices will shift 3 places down the y\begin{align*}y-\end{align*}axis. That means that the ordered pairs for the new vertices will change. Specifically, the y\begin{align*}y-\end{align*}coordinate in each pair will decrease by 3.

Let’s see why this happens.

We can see the change in all of the y\begin{align*}y-\end{align*}coordinates. Compare the top points. The y\begin{align*}y-\end{align*}coordinate on the left is 2. The y\begin{align*}y-\end{align*}coordinate for the corresponding point in the triangle after it moves is -1. The y\begin{align*}y-\end{align*}coordinate decreased by 3. Now compare the left-hand point of each triangle. The y\begin{align*}y-\end{align*}coordinate originally is -2, and the y\begin{align*}y-\end{align*}coordinate after the translation is -5. Again, the difference shows a change of -3 in the y\begin{align*}y-\end{align*}coordinate. For the last point, the y\begin{align*}y-\end{align*}coordinate starts out as -6, and shifts to -9 after the downward slide. For each point, then, the y\begin{align*}y-\end{align*}coordinate decreases by 3 while the x\begin{align*}x-\end{align*}coordinates stay the same. This means that we slid the triangle down 3 places.

We can translate figures in other ways, too. As you might guess, we move figures right or left on the coordinate grid by their x\begin{align*}x-\end{align*}coordinates. We can also move figures diagonally by changing both their x\begin{align*}x-\end{align*} and y\begin{align*}y-\end{align*}coordinates. One way to recognize translations, then, is to compare their points. The x\begin{align*}x-\end{align*}coordinates will all change the same way, and the y\begin{align*}y-\end{align*}coordinates will all change the same way.

To graph a translation, we perform the same change for each point.

We can also identify a reflection by the changes in its coordinates. Recall that in a reflection, the figure flips across a line to make a mirror image of itself. Take a look at the reflection below.

We usually reflect a figure across either the x\begin{align*}x-\end{align*} or the y\begin{align*}y-\end{align*}axis. In this case, we reflected the figure across the y\begin{align*}y-\end{align*}axis. If we compare the figures in the first example vertex by vertex, we see that the x\begin{align*}x-\end{align*}coordinates change but the y\begin{align*}y-\end{align*}coordinates stay the same. This is because the reflection happens from left to right across the y\begin{align*}y-\end{align*}axis. When we reflect across the x\begin{align*}x-\end{align*}axis, the y\begin{align*}y-\end{align*}coordinates change and the x\begin{align*}x-\end{align*}coordinates stay the same. Take a look at this example.

Now let’s compare some of the vertices. In the figure above the coordinates for the upper-left vertex of the original figure are (-5, 5). After we reflect it across the x\begin{align*}x-\end{align*}axis, the coordinates for the corresponding vertex are (-5, -5). How about the lower-right vertex? It starts out at (-1, 1), and after the flip it is at (-1, -1). As you can see, the x\begin{align*}x-\end{align*}coordinates stay the same while the y\begin{align*}y-\end{align*}coordinates change. In fact, the y\begin{align*}y-\end{align*}coordinates all become the opposite integers of the original y\begin{align*}y-\end{align*}coordinates. This indicates that this is a vertical (up/down) reflection or we could say a reflection over the x\begin{align*}x-\end{align*}axis.

In a horizontal (left/right) reflection or a reflection over the y\begin{align*}y-\end{align*}axis, the x\begin{align*}x-\end{align*}coordinates would become integer opposites. Let’s see how.

This is a reflection across the y\begin{align*}y-\end{align*}axis. Compare the points. Notice that the y\begin{align*}y-\end{align*}coordinates stay the same. The x\begin{align*}x-\end{align*}coordinates become the integer opposites of the original x\begin{align*}x-\end{align*}coordinates. Look at the top point of the triangle, for example. The coordinates of the original point are (-4, 6), and the coordinates of the new point are (4, 6). The x\begin{align*}x-\end{align*}coordinate has switched from -4 to 4.

We can recognize reflections by these changes to the x\begin{align*}x-\end{align*} and y\begin{align*}y-\end{align*}coordinates. If we reflect across the x\begin{align*}x-\end{align*}axis, the y\begin{align*}y-\end{align*}coordinates will become opposite. If we reflect across the y\begin{align*}y-\end{align*}axis, the x\begin{align*}x-\end{align*}coordinates will become opposite.

We can also use this information to graph reflections. To graph a reflection, we need to decide whether the reflection will be across the x\begin{align*}x-\end{align*}axis or the y\begin{align*}y-\end{align*}axis, and then change either the x\begin{align*}x-\end{align*} or y\begin{align*}y-\end{align*}coordinates.

Now let’s look at the third kind of transformation: rotations. A rotation is a transformation that turns the figure in either a clockwise or counterclockwise direction. The figure below has been rotated. What are its new coordinates?

The new coordinates of the rectangle’s vertices are (1, -3), (1, 2), (3, 2), and (3, -3). As you can see, both the x\begin{align*}x-\end{align*} and y\begin{align*}y-\end{align*}coordinates changed. Unlike a translation or reflection, a rotation can change both of the coordinates in an ordered pair. Now look closely. One of the points remained exactly the same! We say that we rotated the figure about this point. Imagine you put your finger on this corner of the rectangle and spun it. That’s what happened in the rotation. The rectangle has been rotated 90\begin{align*}90^{\circ}\end{align*} clockwise.

How do we graph a rotation?

When we graph a rotation, we first need to know how much the figure will be rotated. Rotating the rectangle above 90\begin{align*}90^{\circ}\end{align*} stands it up on end. Rotating it 180\begin{align*}180^{\circ}\end{align*} would make it flat again. We also need to know which point we will rotate it around. This is the point that stays the same.

Next, we need to count how many units long and wide the figure is. The figure above stretches from 1 on the x\begin{align*}x-\end{align*}axis to -4 on the x\begin{align*}x-\end{align*}axis. This is a total of 5 units along the x\begin{align*}x-\end{align*}axis. When we rotate a figure 90\begin{align*}90^{\circ}\end{align*}, the distance on the x\begin{align*}x-\end{align*}axis becomes the distance on the y\begin{align*}y-\end{align*}axis. Look at the rectangle. The long sides are horizontal at first, but after we rotate it, they become the vertical sides. This means that the x\begin{align*}x-\end{align*}distance of 5 will become a y\begin{align*}y-\end{align*}distance of 5.

Now, remember the point (1, -3) stays the same, so it is one corner of the rotated figure. We add 5 to the y\begin{align*}y-\end{align*}coordinate to find the next vertex of the rectangle. 3+5=2\begin{align*}-3 + 5 = 2\end{align*}. This puts a vertex at (1, 2).

To find the other points of the rotated rectangle, we need to think about its width. Find the width, or short side, of the original rectangle by counting the units between vertices along the y\begin{align*}y-\end{align*}axis. The rectangle covers 2 units on the y\begin{align*}y-\end{align*}axis. As you might guess, this becomes the x\begin{align*}x-\end{align*}distance in the rotated figure. In other words, we add 2 to the x\begin{align*}x-\end{align*}coordinate of the point that stays the same. 1+2=3\begin{align*}1 + 2 = 3\end{align*}, so another vertex of the rectangle will be (3, -3). To find the fourth and final vertex, add 2 to the x\begin{align*}x-\end{align*}coordinate of the other ordered pair we know, (1, 2). This puts the last vertex at (3, 2).

Write each set of coordinates to show a reflection in the y\begin{align*}y-\end{align*}axis.

Example A

(-3, 1) (0, 3) (1, 2)

Solution: (3, -1)(0, -3)(-1, -2)

Example B

(-3, 6) (-2, 3) (2, 3) (3, 6)

Solution: (3, 6)(2, -3)(-2, -3)(-3,-6)

Example C

True or false. When there is a reflection in the y\begin{align*}y-\end{align*}axis, both coordinates change to opposites.

Solution: False

Here is the original problem once again.

While at the art museum, Tanya found an area where you could create pieces of art using a computer. These graphics could be created and manipulated using a program. She used a computer to create this image.

Tanya has created an image that started in the blue position and moved to the red position.

Do you know what these two figures are called?

When looking at these figures you can see that they are not reflected. A reflection would involve a complete flip over the x\begin{align*}x\end{align*} or y\begin{align*}y\end{align*} axis.

This is not a rotation because the position of the figure does not shift in circular degrees.

This is a translation or a slide. You can see that the figure simply "slides" into its new position.

Guided Practice

Here is one for you to try on your own.

Draw a reflection of the figure below across the x\begin{align*}x-\end{align*}axis.

We need to reflect the rectangle across the x\begin{align*}x-\end{align*}axis, so the “flip” will move the rectangle down. Because the reflection is across the x\begin{align*}x-\end{align*}axis, we’ll need to change the coordinates (which determine where points are up and down).

Specifically, we need to change them to their integer opposites. An integer is the same number with the opposite sign. This gives us the new points.

(3,6)(5,6)(3,1)  (5,1)(3,6)(5,6)(3,1)(5,1)

Now we graph the new points.

Here is the completed reflection.

Explore More

Directions: Use this figure to answer each question. Be sure to write everything in coordinate notation when possible.

1. Translate this figure three units up.

2. Translate this figure four units to the right.

3. Translate this figure five units down.

4. Translate this figure six units to the left.

5. Translate this figure one unit down and two units to the right.

6. Translate this figure two units up and one unit to the left.

7. Translate this figure three units up and one unit to the right.

8. Rotate this figure 180\begin{align*}180\end{align*} degrees.

9. Rotate this figure 90\begin{align*}90\end{align*} degrees.

10. Reflect this figure over the x\begin{align*}x\end{align*} axis.

11. Reflect this figure over the y\begin{align*}y\end{align*} axis.

12. Translate this figure five units up and three units to the right.

13. Translate this figure six units down and four units to the left.

14. True or false. The figure below is an image of a reflection.

15. True or false. This figure has been rotated 180\begin{align*}180^{\circ}\end{align*}.

Vocabulary Language: English

$x-$axis

$x-$axis

The $x-$axis is the horizontal axis in the coordinate plane, commonly representing the value of the input or independent variable.
$y-$axis

$y-$axis

The $y-$axis is the vertical number line of the Cartesian plane.
Coordinate Notation

Coordinate Notation

A coordinate point is the description of a location on the coordinate plane. Coordinate points are commonly written in the form (x, y) where x is the horizontal distance from the origin, and y is the vertical distance from the origin.
Coordinate Plane

Coordinate Plane

The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin. The coordinate plane is also called a Cartesian Plane.
Reflection

Reflection

A reflection is a transformation that flips a figure on the coordinate plane across a given line without changing the shape or size of the figure.
Rotation

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

Transformation

A transformation moves a figure in some way on the coordinate plane.
Translation

Translation

A translation is a transformation that slides a figure on the coordinate plane without changing its shape, size, or orientation.

Date Created:

Nov 30, 2012

Sep 23, 2015
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