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8.18: Transformation Classification

Difficulty Level: At Grade Created by: CK-12
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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.

(4,1)(5,2)(5,0)(6,1)

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

In the last few Concepts, you learned 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 x\begin{align*}x-\end{align*}axis, and a vertical axis, called the y\begin{align*}y-\end{align*}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 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. 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 x\begin{align*}x-\end{align*}coordinates for each point will change but the y\begin{align*}y-\end{align*}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 x\begin{align*}x-\end{align*}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.

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

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 ABC\begin{align*}ABC\end{align*} (0, 1)(1, 3)(4, 0) translate this figure up 4.

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

Example B

Triangle DEF\begin{align*}DEF\end{align*} (-3, 2)(1, 6)(2, 1) translate this figure down 2.

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

Example C

Triangle DEF\begin{align*}DEF\end{align*} (-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.

(4,1)(5,2)(5,0)(6,1)

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 y\begin{align*}y-\end{align*}axis. Therefore, the x\begin{align*}x-\end{align*}coordinate is going to change and become negative in each of the four vertices of the diamond. Here are the coordinates.

(4,1)(5,0)(5,2)(6,1)

Next, we can reflect the original diamond in the first quadrant over the x\begin{align*}x-\end{align*}axis into the fourth quadrant. Here the y\begin{align*}y-\end{align*}coordinates will be negative.

(4,1)(5,0)(5,2)(6,1)

Finally we can reflect this diamond over the y\begin{align*}y-\end{align*}axis into the third quadrant. Notice that here the x\begin{align*}x\end{align*} and y\begin{align*}y \end{align*}-coordinates will both be negative.

(4,1)(5,0)(5,2)(6,1)

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 X\begin{align*}X\end{align*} that crosses through the original pattern could you do it? Explain your thinking with a friend and then add in the X\begin{align*}X\end{align*} to the coordinate grid with the diamonds.

Vocabulary

Here are the vocabulary words found in this Concept.

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 x\begin{align*}x-\end{align*} axis and a y\begin{align*}y-\end{align*} axis.
x\begin{align*}x-\end{align*}axis
the horizontal line in a coordinate plane.
y\begin{align*}y-\end{align*}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.

This time we need to perform two movements, both left and up. That means we will change both the x\begin{align*}x-\end{align*} and y\begin{align*}y-\end{align*}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 y\begin{align*}y-\end{align*}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 x\begin{align*}x-\end{align*} and y\begin{align*}y-\end{align*}coordinates in the ordered pairs and then comparing these to the points we graphed. This time we subtract 4 from each x\begin{align*}x-\end{align*}coordinate (because we moved left; imagine a number line) and add 2 to each y\begin{align*}y-\end{align*}coordinate. Let’s see what happens.

(3, 2)(4,2)(1,4)4+24+2  4+2(1, 4)   (0, 0) (3,2)

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

Video Review

Here is a video for review.

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 DEF\begin{align*}DEF\end{align*} (-1, 2)(1, 6)(2, 1)

11. Triangle DEF\begin{align*}DEF\end{align*} (-3, 2)(1, 6)(2, 1)

12. Triangle DEF\begin{align*}DEF\end{align*} (0, 2)(1, 6)(2, 1)

13. Triangle DEF\begin{align*}DEF\end{align*} (4, -2)(1, 6)(2, 1)

14. Triangle DEF\begin{align*}DEF\end{align*} (5, 3)(1, 6)(2, 1)

15. Triangle DEF\begin{align*}DEF\end{align*} (4, 4)(1, 6)(2, 1)

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 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.
Transformation

Transformation

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

Date Created:

Nov 30, 2012

Jan 26, 2016
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