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8.19: Translations, Rotations, and Reflections

Difficulty Level: At Grade Created by: CK-12

Let's Think About It

License: CC BY-NC 3.0

Tyler takes a picture of an item and its reflection. He places a coordinate plane over the picture. The coordinate plane is positioned so that the xaxis separates the image from the reflection. He then makes the grid according to the key features of the picture, so that a point at (2, 0) is reflected at the point (-2, 0). If the original coordinates of the image are (3, 0), (4, 6) and (5, 1), what are the coordinates of the reflection?

In this concept, you will learn how to find the coordinates for translations, rotations and reflections.

Guidance

When you perform translations, you 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.

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

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

Figures are usually reflected across either the x or the yaxis. In this case, the figure is reflected across the yaxis. If you compare the figures in the first example vertex by vertex, you see that the xcoordinates change but the ycoordinates stay the same. This is because the reflection happens from left to right across the yaxis. When you reflect across the xaxis, the ycoordinates change and the xcoordinates stay the same. Take a look at this example.

In the figure above the coordinates for the upper-left vertex of the original figure are (-5, 5). After you reflect it across the xaxis, 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 xcoordinates stay the same while the ycoordinates change. In fact, the ycoordinates all become the opposite integers of the original ycoordinates. This indicates that this is a vertical (up/down) reflection or a reflection over the xaxis.

In a horizontal (left/right) reflection or a reflection over the yaxis, the xcoordinates would become integer opposites. Let’s look at an example.

This is a reflection across the yaxis. Compare the points. Notice that the ycoordinates stay the same. The xcoordinates become the integer opposites of the original xcoordinates. 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 xcoordinate has switched from -4 to 4.

You can recognize reflections by these changes to the x and ycoordinates. If you reflect across the xaxis, the ycoordinates will become opposite. If you reflect across the yaxis, the xcoordinates will become opposite.

You can also use this information to graph reflections. To graph a reflection, you need to decide whether the reflection will be across the xaxis or the yaxis, and then change either the x or ycoordinates.

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 and ycoordinates changed. One of the points remained exactly the same. That means the figure was rotated 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 clockwise.

When you graph a rotation, you first need to know how much the figure will be rotated. Rotating the rectangle above 90 stands it up on end. Rotating it 180 would make it flat again. You also need to know which point you will rotate it around. This is the point that stays the same.

Next, you need to count how many units long and wide the figure is. The figure above stretches from 1 on the xaxis to -4 on the xaxis. This is a total of 5 units along the xaxis. When you rotate a figure 90, the distance on the xaxis becomes the distance on the yaxis. Look at the rectangle. The long sides are horizontal at first, but after you rotate it, they become the vertical sides. This means that the xdistance of 5 will become a ydistance of 5.

Now, remember the point (1, -3) stays the same, so it is one corner of the rotated figure. You add 5 to the ycoordinate to find the next vertex of the rectangle. 3+5=2. This puts a vertex at (1, 2).

To find the other points of the rotated rectangle, you need to think about its width. Find the width, or short side, of the original rectangle by counting the units between vertices along the yaxis. The rectangle covers 2 units on the yaxis. As you might guess, this becomes the xdistance in the rotated figure. In other words, you add 2 to the xcoordinate of the point that stays the same. 1+2=3, so another vertex of the rectangle will be (3, -3). To find the fourth and final vertex, add 2 to the xcoordinate of the other ordered pair, (1, 2). This puts the last vertex at (3, 2).

Write each set of coordinates to show a reflection in the yaxis.

Guided Practice

The figure below is reflected across the xaxis. What are the coordinates of the reflection?

First, remember how to reflect across the xaxis.

Flip the shape across the xaxis.

Next, remember the rules.

The  ycoordinates will change to their integer opposites.

Then, write the new points.

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

The answer is that the coordinates of the reflection are (3, -6), (3, -1), (5, -6), (5, -1).

Examples

Example 1

Determine if the change is the result of a translation, reflection or rotation.

After a transformation, the vertices (3, 1), (0, 3), (1, 2) become (3, -1), (0, -3), (1, -2).

First, look at the coordinates to see which coordinate changes.

The xcoordinates stay the same and the ycoordinates change for each of the points.

Next, determine the type of change for the coordinates.

The new ycoordinates are opposites of the original coordinates.

Then, state the type of transformation.

Reflection

The answer is that the change is the result of a reflection.

Example 2

Determine if the change is the result of a translation, reflection or rotation.

After a transformation, the vertices (-3, 6), (-2, 3), (2, 3), (3, 6) become (-3, -1), (-2, 8), (2, 8), (3, 11).

First, look at the coordinates to see which coordinate changes.

The xcoordinates stay the same and the ycoordinates change for each of the points.

Next, determine the type of change for the coordinates.

The new ycoordinates are each 5 more than the original coordinates.

Then, state the type of transformation.

Translation

The answer is that the change is the result of a translation.

Example 3

The points (1, 2), (3, 7), (3, 4), (5, 6) are reflected over the yaxis. What are the new coordinates?

First, remember the changes that occur with a reflection over the yaxis.

The xcoordinates become the opposite and the ycoordinates stay the same.

Then, write the new coordinates.

(-1, 2), (-3, 7), (-3, 4), (-5, 6)

The answer is that the new coordinates are (-1, 2), (-3, 7), (-3, 4), (-5, 6).

True or false. When there is a reflection in the yaxis, both coordinates change to opposites.

Follow Up

License: CC BY-NC 3.0

Remember Tyler and his picture? If the original coordinates of the image are (3, 0), (4, 6) and (5, 1), what are the coordinates of the reflection?

First, remember how to reflect across the xaxis.

Flip the shape across the xaxis.

Next, remember the rules.

The  ycoordinates will change to their integer opposites.

Then, write the new points.

(3, 0), (4, -6), (5, -1)

The answer is that the coordinates of the reflection are (3, 0), (4, -6), (5, -1).

Video Review

Explore More

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 degrees.

9. Rotate this figure 90 degrees.

10. Reflect this figure over the x axis.

11. Reflect this figure over the y 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.

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 8.19. 

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0

Description

Difficulty Level:

At Grade

Grades:

Date Created:

Dec 02, 2015

Last Modified:

Dec 02, 2015
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