# 6.14: Recognizing Translation Transformations

**At Grade**Created by: CK-12

**Practice**Geometric Translations

Have you ever seen an image like this? This diagram represents a transformation. Do you know which one?

In this concept, you will learn to recognize translation transformations.

### Translation

A **transformation** is the movement of a geometric figure on the coordinate plane. There are several different types of transformations. One of these is called a translation. A **translation** is when a geometric figure slides up, down, left or right on the coordinate plane. The figure moves its location, but doesn’t change its orientation. It also doesn’t change its size or shape.

When you perform translations, you slide a figure left or right, up or down. This means that, in the coordinate plane, the coordinates for the vertices of the figure will change.

You can represent this triangle by using coordinate notation. Coordinate notation is when you write ordered pairs to represent each of the vertices of a geometric figure that has been graphed on the coordinate plane.

\begin{align*}\begin{array}{rcl} (-1, 5) \\ (-1, 2) \\ (-5, 2) \end{array}\end{align*}

These are the coordinates of the vertices of the triangle.

If you slide this triangle 3 places down, all of its vertices will shift 3 places down the \begin{align*}y\end{align*}-axis. That means that the ordered pairs for the new vertices will change. Specifically, the \begin{align*}y\end{align*}-coordinate in each pair will decrease by 3.

Now the \begin{align*}y\end{align*}-coordinate of each ordered pair decreased by three units. You can see how the ordered pairs changed from the first image to the next image.

\begin{align*}\begin{array}{rcl} (-1, 5) & \rightarrow \quad \ \ (-1, 2) \\ (-1, 2) & \rightarrow \quad (-1, -1) \\ (-5, 2) & \rightarrow \ \quad (-5, -1) \end{array}\end{align*}

The \begin{align*}y\end{align*}-coordinate changed from 5 to 2, from -1 to 2 and from 2 to -1. As you move down, the value of the coordinate also moved down. Similarly, if you were to move the image up three units on the \begin{align*}y\end{align*}-axis, then you would increase the value of the \begin{align*}y\end{align*}-coordinate by three units. Also if you were to move the image to the right then you would increase the \begin{align*}x\end{align*}-coordinate. If you were to move it to the left, then you would decrease the \begin{align*}x\end{align*}-coordinate.

You can translate figures in other ways too. You can move figures diagonally by changing both their \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates. One way to recognize translations, then, is to compare their points. The \begin{align*}x\end{align*}-coordinates will all change the same way, and the \begin{align*}y\end{align*}-coordinates will all change the same way. To graph a translation, you perform the same change for each point.

Now let’s try graphing a translation. Take a look at this one.

Graph the following translation five places to the right.

First, you can see by looking at this square that there are four vertices, so there are four sets of ordered pairs to represent these vertices. List the ordered pairs.

\begin{align*}\begin{array}{rcl} (-4, 3) \\ (-1, 3) \\ (-1, 6) \\ (-4, 6) \end{array}\end{align*}

Next, the translation is to move the square five places to the right. That means that you are going to change the \begin{align*}x\end{align*}-coordinate and not the \begin{align*}y\end{align*}-coordinate. List the new ordered pairs.

\begin{align*}\begin{array}{rcl} (-4 + 5, 3) &=& (1, 3) \\ (-1 + 5, 3) &=& (4, 3) \\ (-1 + 5, 6) &=& (4, 6) \\ (-4 + 5 ,6) &=& (1, 6) \end{array}\end{align*}

Then, let’s look at the graph of the translation.

This is the answer.

Notice that while it is helpful to graph the square both first and then as a translation, it isn’t necessary to do so to figure out the coordinate notation. If you know the vertices of the figure that you are translating and you know how you are moving it, then you can figure out the new coordinates of the vertices.

### Examples

#### Example 1

Looking at this diagram, you can see that the figure, a quadrilateral has been shifted to the right and then up. It has not been flipped or turned. It has been moved, so this is a slide. Another name for a slide is a translation.

The answer is that this transformation is a slide.

#### Example 2

A triangle with the coordinates (0, 2), (2, 2) and (2, 5) is graphed on the coordinate grid. Find the coordinates of a translation moved three units down. Then graph the translation.

First, notice the direction of the translation. It is to move the triangle three units down. Down means that you will be subtracting three and down also means that you will be changing the \begin{align*}y\end{align*}-coordinate since up and down involves the \begin{align*}y\end{align*}-axis. List the ordered pairs for the translated triangle.

\begin{align*}\begin{array}{rcl} (0, 2-3) &=& (0, -1) \\ (2, 2-3) &=& (2, -1) \\ (2, 5-3) &=& (2, 2) \end{array}\end{align*}

Next, graph the translation.

This is the answer.

Use this diagram to answer the following questions.

#### Example 3

Is this figure a translation?

Look at \begin{align*}\triangle IRT\end{align*} and \begin{align*}\triangle I^{\prime} R^{\prime} T^{\prime}\end{align*}. What is happening to the triangles?

The vertices of \begin{align*}\triangle IRT\end{align*} have moved to the left 6 units and down 4 units. \begin{align*}\triangle I^{\prime} R^{\prime} T^{\prime}\end{align*} is the translation of \begin{align*}\triangle IRT\end{align*}.

The answer is that yes, this is a translation. Remember a translation is a transformation that is informally called a slide or a glide.

#### Example 4

How many units up or down has the figure been moved?

First, look at the vertices in the triangles to compare.

Point \begin{align*}I\end{align*} has coordinates (-2, 8) and Point \begin{align*}I^{\prime}\end{align*} has coordinates (4, 4). All other vertices change in the same way.

Next, look at the change in the coordinates. Asking if the figure means up or down is the same as asking what the change in the \begin{align*}y\end{align*}-coordinate is

\begin{align*}8 \ {\color{red}- \ 4} = 4\\\end{align*}

The answer is that the figure moves 4 units down or -4.

#### Example 5

How many units to the right or left?

First, look at the vertices in the triangles to compare.

Point \begin{align*}I\end{align*} has coordinates (-2, 8) and Point \begin{align*}I^{\prime}\end{align*} has coordinates (4, 4). All other vertices change in the same way.

Next, look at the change in the coordinates. Asking if the figure means left or right is the same as asking what the change in the \begin{align*}x\end{align*}-coordinate is

\begin{align*}-2 \ {\color{red}+ \ 6} = 4\end{align*}

The answer is that the figure moves 6 units to the right or +6.

### Review

Use the following diagram to answer each question.

1. What kind of transformation is shown in the diagram?

2. What are the coordinates of the first triangle?

3. What are the coordinates of the translated triangle?

4. What direction was the triangle first moved?

5. How many units?

6. Then which direction was the triangle moved?

7. How many units?

8. What are the coordinates of the first triangle?

9. What are the coordinates of the translated triangle?

10. Was the figure moved right or left?

11. How many units?

12. Was the figure moved up or down?

13. How many units?

14. True or false. Another name for a slide is a translation.

15. True or false. A rotation and a translation have the same characteristics.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.14.

### Resources

Center of Rotation

In a rotation, the center of rotation is the point that does not move. The rest of the plane rotates around this fixed point.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.

Image

The image is the final appearance of a figure after a transformation operation.Preimage

The pre-image is the original appearance of a figure in a transformation operation.Reflections

Reflections are transformations that result in a "mirror image" of a parent function. They are caused by differing signs between parent and child functions.Transformation

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

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

In this concept, you will learn to recognize translation transformations also called slides.

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