In one room of the museum was a King’s bedroom. The furniture in the room was large and wooden and old with great golden cloths. On the walls was a beautiful red, blue and gold pattern.

Jessica thought that the pattern was the most beautiful one that she had ever heard.

“I love this,” she said to Mrs. Gilman. “I want to draw it, but I’m not sure how.”

“Well, you could break it up into a coordinate grid since the pattern repeats itself and use what we have learned about transformations to draw it in.”

“How could I get started?” Jessica asked.

“Well, start by drawing the coordinate grid, then use these coordinates for one of the diamonds. See if you can figure it out from there.”

In Jessica’s notebook, Mrs. Gilman wrote down the following coordinates.

Jessica began to draw it in. Then she got stuck.

**
This is where you come in. This Concept will teach you all about drawing transformations. Follow along closely and you can help Jessica draw in the diamonds at the end of the Concept in each quadrant.
**

### Guidance

Previously we worked on how to identify and perform different
**
transformations.
**
Remember that a

**transformation is when we move a figure in some way, even though we don’t change the figure at all.**This Concept will teach you how to identify and perform transformations in the coordinate plane.

**
The
**
**
coordinate plane
**

**is a representation of two-dimensional space. It has a horizontal axis, called the**

*axis***, and a vertical axis, called the**

*axis***.**We can graph and move geometric figures on the coordinate plane.

**
Do you remember the three types of transformations?
**

The first is a
**
translation
**
or slide. A translation

**moves a figure up, down, to the right, to the left or diagonal without altering the figure.**

The second is a
**
reflection
**
or flip. A reflection

**makes a mirror image of the figure over a line of symmetry.**The line of symmetry can be vertical or horizontal.

The third is a
**
rotation
**
or turn. A rotation

**moves a figure in a circle either clockwise or counterclockwise.**

**
Now let’s look at performing each type of transformation in the coordinate plane.
**

As we have said, 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 example 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 axis. That means that the ordered pairs for the new vertices will change. Specifically, the coordinate in each pair will decrease by 3.

Let’s see why this happens.

**
We can see the change in all of the
coordinates.
**
Compare the top points. The
coordinate on the left is 2. The
coordinate for the corresponding point in the triangle after it moves is -1. The
coordinate decreased by 3. Now compare the left-hand point of each triangle. The
coordinate originally is -2, and the
coordinate after the translation is -5. Again, the difference shows a change of -3 in the
coordinate. For the last point, the
coordinate starts out as -6, and shifts to -9 after the downward slide. For each point, then, the
coordinate decreases by 3 while the
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
coordinates. We can also move figures diagonally by changing both their
and
coordinates.
**
One way to recognize translations, then, is to compare their points. The
coordinates will all change the same way, and the
coordinates will all change the same way.

**
To graph a translation, we perform the same change for each point.
**
Let’s try graphing a translation.

Slide the following figure 5 places to the right.

**
In this translation, we will move the figure to the right. That means the
coordinates for each point will change but the
coordinates will not. We simply count 5 places to the right from each point and make a new point.
**

**
Once we relocate each point 5 places to the right, we can connect them to make the new figure that shows the translation.
**

We can check to see if we performed the translation correctly by adding 5 to each
coordinate (because we moved to the right) and then checking these against the ordered pairs of the figure you drew. This is called
**
coordinate notation.
**
Notice that each point is represented by coordinates.

**
These are the points we graphed, so we have performed the translation correctly.
**

Use coordinate notation to write the coordinates of each translated triangle. The vertices of the original figure have been given to you.

#### Example A

Triangle (0, 1)(1, 3)(4, 0) translate this figure up 4.

**
Solution: (0, 5)(1, 7)(4, 4)
**

#### Example B

Triangle (-3, 2)(1, 6)(2, 1) translate this figure down 2.

**
Solution: (-3,0)(1, 4) (2, -1)
**

#### Example C

Triangle (-3, 2)(1, 6)(2, 1) translate this figure to the right 3.

**
Solution: (0, 2)(4, 6)(5, 1)
**

Here is the original problem once again. Reread it before working on the drawing.

In one room of the museum was a King’s bedroom. The furniture in the room was large and wooden and old with great golden cloths. On the walls was a beautiful red, blue and gold pattern.

Jessica thought that the pattern was the most beautiful one that she had ever heard.

“I love this,” she said to Mrs. Gilman. “I want to draw it, but I’m not sure how.”

“Well, you could break it up into a coordinate grid since the pattern repeats itself and use what we have learned about transformations to draw it in.”

“How could I get started?” Jessica asked.

“Well, start by drawing the coordinate grid, then use these coordinates for one of the diamonds. See if you can figure it out from there.”

In Jessica’s notebook, Mrs. Gilman wrote down the following coordinates.

Jessica began to draw it in. Then she got stuck.

**
Now we can draw this diamond in on a coordinate grid. It belongs in Quadrant one. Now we want to draw a diamond into each of the other three quadrants. We can draw this, but we can also use mathematics to figure out the coordinates for each of the other diamonds first.
**

**
The diamond in the second quadrant is reflected over the
axis. Therefore, the
coordinate is going to change and become negative in each of the four vertices of the diamond. Here are the coordinates.
**

**
Next, we can reflect the original diamond in the first quadrant over the
axis into the fourth quadrant. Here the
coordinates will be negative.
**

**
Finally we can reflect this diamond over the
axis into the third quadrant. Notice that here the
and
-coordinates will both be negative.
**

**
Did you notice any patterns? Take a minute and create this pattern of diamonds in a coordinate grid. Then you will have an even deeper understanding of how a pattern like this one is created.
**

**
If you wanted to add in the gold
that crosses through the original pattern could you do it? Explain your thinking with a friend and then add in the
to the coordinate grid with the diamonds.
**

### Vocabulary

- Transformation
- a figure that is moved in the coordinate grid is called a transformation.

- Coordinate Plane
- a representation of a two-dimensional plane using an axis and a axis.

- axis
- the horizontal line in a coordinate plane.

- axis
- the vertical line in a coordinate plane.

### Guided Practice

Here is one for you to try on your own.

Slide the following figure 4 places to the left and 2 places up.

**
Answer
**

This time we need to perform two movements, both left and up. That means we will change both the and coordinates of the ordered pairs. We graph each point by counting 4 places to the left first, and from there 2 places up (2 places up from where you started, not 2 places up from the axis!). Make a mark and repeat this process for each point. Then connect the new points.

Again, we can check that we performed the translation correctly by changing the and coordinates in the ordered pairs and then comparing these to the points we graphed. This time we subtract 4 from each coordinate (because we moved left; imagine a number line) and add 2 to each coordinate. Let’s see what happens.

**
These are the points we graphed, so we performed the translation correctly.
**

### Video Review

This is a James Sousa video that reviews the coordinate plane.

### Practice

Directions: Identify the transformations shown below as a translation, reflection, or rotation.

9. True or false. This figure has been translated 5 places to the right.

Directions : Translate each figure to the right 6 places and up 1.

10. Triangle (-1, 2)(1, 6)(2, 1)

11. Triangle (-3, 2)(1, 6)(2, 1)

12. Triangle (0, 2)(1, 6)(2, 1)

13. Triangle (4, -2)(1, 6)(2, 1)

14. Triangle (5, 3)(1, 6)(2, 1)

15. Triangle (4, 4)(1, 6)(2, 1)