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8.18: Transformation Classification

Difficulty Level: At Grade Created by: CK-12
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Let's Think About It

Mrs. Gilcrest has designed a special version of the game Battleship. Her version requires students to use coordinates such as (1, 2) to guess the location of a ship. The ships are strategically placed so that each is a translation of the other. In order to keep the game easier for the students, she translated the ships the same distance up or down, right or left. If the first ship is at (3, 6), (3, 7), (3, 8) and the second ship is translated 4 units to the right and 4 units up, what are the coordinates of the second ship?

In this concept, you will learn how to classify transformations.

Guidance

The coordinate plane is a representation of two-dimensional space. It has a horizontal axis, called the xaxis, and a vertical axis, called the yaxis. You can graph and move geometric figures on the coordinate plane.

Remember the three types of transformations: translation, reflection and rotation.

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

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. Take a look at the example below.

Now let’s look at performing a translation or slide of this figure.

You can choose the number places that you want to move the triangle and the direction that you wish to move it in. If you slide this triangle 3 places down, all of its vertices will shift 3 places down the yaxis and the ycoordinate in each pair will decrease by 3.

Let’s see why this happens.

You can see the change in all of the ycoordinates. Compare the top points. The ycoordinate on the left is 2. The ycoordinate for the corresponding point in the triangle after it moves is -1. The ycoordinate decreases by 3. Now compare the left-hand point of each triangle. The ycoordinate originally is -2, and the ycoordinate after the translation is -5. Again, the difference shows a change of -3 in the ycoordinate. For the last point, the ycoordinate starts out as -6, and shifts to -9 after the downward slide. For each point, the ycoordinate decreases by 3 while the xcoordinates stay the same.

You can translate figures in other ways, too. As you might guess, you move figures right or left on the coordinate grid by their xcoordinates. You can also move figures diagonally by changing both their x and ycoordinates. One way to recognize translations, then, is to compare their points. The xcoordinates will all change the same way, and the ycoordinates will all change the same way.

Here is an example of how to graph a translation.

Slide the following figure 5 places to the right.

In this translation, you will move the figure to the right. That means the xcoordinates for each point will change but the ycoordinates will not. You simply count 5 places to the right from each point and make a new point.

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

You can check to see if you performed the translation correctly by adding 5 to each xcoordinate (because you 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 you graphed, so you have performed the translation correctly.

Guided Practice

Solve this problem.

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

First, graph the new points.

Graph each point by counting 4 places to the left, and from there 2 places up.

Then, form the new triangle.

Connect the new points.

You can check the translation by changing the x and ycoordinates in the ordered pairs and then comparing these to the points you graphed. This time subtract 4 from each xcoordinate and add 2 to each ycoordinate. Let’s see what happens.

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

Examples

Example 1

Translate triangle ABC (0, 1), (1, 3), (4, 0) up 4.

First, remember whether up is a move on the xaxis or the yaxis.

Up is a move on the yaxis.

Next, add 4 to each of the yvalues.

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

Then, write the new vertices.

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

The new triangle has coordinates (0, 5), (1, 7), (4, 4).

Example 2

Translate triangle DEF (-3, 2),(1, 6),(2, 1) down 2.

First, remember whether down is a move on the xaxis or the yaxis.

Down is a move on the yaxis.

Next, subtract 2 from each of the yvalues.

(-3, 2-2), (1, 6-2), (2, 1-2)

Then, write the new vertices.

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

The new triangle has coordinates (-3, 0), (1, 4), (2, -1)

Example 3

Translate triangle XYZ (-5, 4), (1, 8), (3, 5) to the right 3.

First, remember whether right is a move on the xaxis or the yaxis.

Right is a move on the xaxis.

Next, add 3 to each xvalue.

(-5+3, 4), (1+3, 8), (3+3, 5)

Then write the new vertices

(-2, 4), (4, 8), (6, 5)

The new triangle has coordinates (-2, 4), (4, 8), (6, 5).

Follow Up

Remember Mrs. Gilcrest and her special version of the game Battleship? She translated the ships the same distance up or down, right or left. If the first ship is at (3, 6), (3, 7), (3, 8) and the second ship is translated 4 units to the right and 4 units up, what are the coordinates of the second ship?

First, remember the signs associated with a move to the right and a move up.

Right is a move in the positive direction on the xaxis and up is a move in the positive direction on the yaxis.

Next, add the moves to the coordinates.

(3+4, 6+4), (3+4, 7+4), (3+4, 8+4)

Then write the new vertices

(7, 10), (7, 11), (7, 12)

The coordinates of the second ship are (7, 10), (7, 11), (7, 12).

Video Review

Explore More

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.

Translate each figure to the right 6 places and up 1. Then write the new coordinates for the figure.

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

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

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

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

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

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

Vocabulary

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.

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Date Created:
Dec 02, 2015
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Mar 23, 2016
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