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# 6.14: Recognizing Translation Transformations

Difficulty Level: At Grade Created by: CK-12
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Practice Geometric Translations

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Have you ever seen something that looked like this? Take a look at this dilemma.

This diagram represents a transformation. Do you know which one?

Pay attention and you will learn how to identify transformations like this one.

### Guidance

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 position. It also doesn't change its size or shape.

Let's look at translations or slides.

When we perform translations, we 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.

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

(-1, 5)

(-1, 2)

(-5, 2)

These are the coordinates of the vertices of the triangle.

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.

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

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

The y\begin{align*}y-\end{align*}coordinate changed from 5 to 2, from -1 to 2 and from 2 to -1. As we move down, the value of the coordinate also moved down.

If we were to move the image up three units on the y\begin{align*}y-\end{align*}axis, then we would increase the value of the y\begin{align*}y-\end{align*}coordinate by three units.

If we were to move the image to the right then we would increase the x\begin{align*}x-\end{align*}coordinate. If we were to move it to the left, then we would decrease the x\begin{align*}x-\end{align*}coordinate.

We can translate figures in other ways too. We can 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.

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

Graph the following translation five places to the right.

Now 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. Here are the ordered pairs.

(-4, 3)

(-1, 3)

(-1, 6)

(-4, 6)

The translation is to move the square five places to the right. That means that we are going to change the x\begin{align*}x-\end{align*}coordinate and not the y\begin{align*}y-\end{align*}coordinate.

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

Now let’s look at the graph of the translation.

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.

Use this diagram to answer the following questions.

#### Example A

Is this figure a translation?

Solution: Yes, it shows a slide.

#### Example B

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

Solution: Four units down or -4

#### Example C

How many units to the right or left?

Solution: Six units to the right or +6

Now let's go back to the dilemma from the beginning of the Concept.

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.

This is our answer.

### Vocabulary

Transformation
moving a geometric figure on the coordinate plane.
Coordinate Notation
using ordered pairs to represent the vertices of a figure that has been graphed on the coordinate plane.
Reflection
A flip of a figure on the coordinate plane.
Translation
A slide – when a figure moves up, down, left or right on the coordinate plane, but does not change position.

### Guided Practice

Here is one for you to try on your own.

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.

Solution

To work through this one, first notice the direction of the translation. It is to move the triangle three units down. Down means that we will be subtracting three and down also means that we will be changing the y\begin{align*}y-\end{align*}coordinate since up and down involves the y\begin{align*}y-\end{align*}axis. Here is our action.

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

If we were to graph the translation, here is what we would see.

### Practice

Directions: 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.

### Notes/Highlights Having trouble? Report an issue.

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### Vocabulary Language: English

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.

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