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8.19: Translations, Rotations, and Reflections

Difficulty Level: At Grade Created by: CK-12
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Have you ever tried to create a piece of art? Look at what Tanya discovered at the art museum.

While at the art museum, Tanya found an area where you could create pieces of art using a computer. These graphics could be created and manipulated using a program. She used a computer to create this image.

Tanya has created an image that started in the blue position and moved to the red position.

Do you know what these two figures are called?

In the coordinate plane, you can change figures in many different ways. This Concept will teach you all about this.


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 image 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 yaxis. That means that the ordered pairs for the new vertices will change. Specifically, the ycoordinate in each pair will decrease by 3.

Let’s see why this happens.

We 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 decreased 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, then, the ycoordinate decreases by 3 while the xcoordinates 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 xcoordinates. We 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.

To graph a translation, we perform the same change for each point.

We can also identify a reflection by the changes in its coordinates. Recall that in a reflection, the figure flips across a line to make a mirror image of itself. Take a look at the reflection below.

We usually reflect a figure across either the x or the yaxis. In this case, we reflected the figure across the xaxis. If we compare the figures in the first example vertex by vertex, we see that the xcoordinates change but the ycoordinates stay the same. This is because the reflection happens from left to right across the xaxis. When we reflect across the yaxis, the ycoordinates change and the xcoordinates stay the same. Take a look at this example.

Now let’s compare some of the vertices. In the figure above the coordinates for the upper-left vertex of the original figure are (-5, 5). After we reflect it across the yaxis, the coordinates for the corresponding vertex are (-5, -5). How about the lower-right vertex? It starts out at (-1, 1), and after the flip it is at (-1, -1). As you can see, the xcoordinates stay the same while the ycoordinates change. In fact, the ycoordinates all become the opposite integers of the original ycoordinates. This indicates that this is a vertical (up/down) reflection or we could say a reflection over the xaxis.

In a horizontal (left/right) reflection or a reflection over the yaxis, the xcoordinates would become integer opposites. Let’s see how.

This is a reflection across the xaxis. Compare the points. Notice that the ycoordinates stay the same. The xcoordinates become the integer opposites of the original xcoordinates. Look at the top point of the triangle, for example. The coordinates of the original point are (-4, 6), and the coordinates of the new point are (4, 6). The xcoordinate has switched from -4 to 4.

We can recognize reflections by these changes to the x and ycoordinates. If we reflect across the xaxis, the xcoordinates will become opposite. If we reflect across the yaxis, the ycoordinates will become opposite.

We can also use this information to graph reflections. To graph a reflection, we need to decide whether the reflection will be across the xaxis or the yaxis, and then change either the x or ycoordinates.

Now let’s look at the third kind of transformation: rotations. A rotation is a transformation that turns the figure in either a clockwise or counterclockwise direction. The figure below has been rotated. What are its new coordinates?

The new coordinates of the rectangle’s vertices are (1, -3), (1, 2), (3, 2), and (3, -3). As you can see, both the x and ycoordinates changed. Unlike a translation or reflection, a rotation can change both of the coordinates in an ordered pair. Now look closely. One of the points remained exactly the same! We say that we rotated the figure about this point. Imagine you put your finger on this corner of the rectangle and spun it. That’s what happened in the rotation. The rectangle has been rotated 90 clockwise.

How do we graph a rotation?

When we graph a rotation, we first need to know how much the figure will be rotated. Rotating the rectangle above 90 stands it up on end. Rotating it 180 would make it flat again. We also need to know which point we will rotate it around. This is the point that stays the same.

Next, we need to count how many units long and wide the figure is. The figure above stretches from 1 on the xaxis to -4 on the xaxis. This is a total of 5 units along the xaxis. When we rotate a figure 90, the distance on the xaxis becomes the distance on the yaxis. Look at the rectangle. The long sides are horizontal at first, but after we rotate it, they become the vertical sides. This means that the xdistance of 5 will become a ydistance of 5.

Now, remember the point (1, -3) stays the same, so it is one corner of the rotated figure. We add 5 to the ycoordinate to find the next vertex of the rectangle. 3+5=2. This puts a vertex at (1, 2).

To find the other points of the rotated rectangle, we need to think about its width. Find the width, or short side, of the original rectangle by counting the units between vertices along the yaxis. The rectangle covers 2 units on the yaxis. As you might guess, this becomes the xdistance in the rotated figure. In other words, we add 2 to the xcoordinate of the point that stays the same. 1+2=3, so another vertex of the rectangle will be (3, -3). To find the fourth and final vertex, add 2 to the xcoordinate of the other ordered pair we know, (1, 2). This puts the last vertex at (3, 2).

Write each set of coordinates to show a reflection in the yaxis.

Example A

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

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

Example B

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

Solution: (3, 6)(2, -3)(-2, -3)(-3,-6)

Example C

True or false. When there is a reflection in the yaxis, both coordinates change to opposites.

Solution: False

Here is the original problem once again.

While at the art museum, Tanya found an area where you could create pieces of art using a computer. These graphics could be created and manipulated using a program. She used a computer to create this image.

Tanya has created an image that started in the blue position and moved to the red position.

Do you know what these two figures are called?

When looking at these figures you can see that they are not reflected. A reflection would involve a complete flip over the x or y axis.

This is not a rotation because the position of the figure does not shift in circular degrees.

This is a translation or a slide. You can see that the figure simply "slides" into its new position.


Here are the vocabulary words that are found in this Concept.

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-axis and a y-axis.
the horizontal line in a coordinate plane.
the vertical line in a coordinate plane.
a slide. A figure is moved up, down, left or right.
a flip. A figure can be flipped over the xaxis or the yaxis.
a turn. A figure can be turned clockwise or counterclockwise.
Coordinate Notation
notation that shows where the figure is located in the coordinate plane. The vertices of the figure are represented using ordered pairs.

Guided Practice

Here is one for you to try on your own.

Draw a reflection of the figure below across the xaxis.


We need to reflect the rectangle across the yaxis, so the “flip” will move the rectangle down. Because the reflection is across the xaxis, we’ll need to change the coordinates (which determine where points are up and down).

Specifically, we need to change them to their integer opposites. An integer is the same number with the opposite sign. This gives us the new points.

(3,6)(5,6)(3,1)  (5,1)(3,6)(5,6)(3,1)(5,1)

Now we graph the new points.

Here is the completed reflection.


Directions: Use this figure to answer each question. Be sure to write everything in coordinate notation when possible.

1. Translate this figure three units up.

2. Translate this figure four units to the right.

3. Translate this figure five units down.

4. Translate this figure six units to the left.

5. Translate this figure one unit down and two units to the right.

6. Translate this figure two units up and one unit to the left.

7. Translate this figure three units up and one unit to the right.

8. Rotate this figure 180 degrees.

9. Rotate this figure 90 degrees.

10. Reflect this figure over the x axis.

11. Reflect this figure over the y axis.

12. Translate this figure five units up and three units to the right.

13. Translate this figure six units down and four units to the left.

14. True or false. The figure below is an image of a reflection.

15. True or false. This figure has been rotated 180.




The x-axis is the horizontal axis in the coordinate plane, commonly representing the value of the input or independent variable.


The y-axis is the vertical number line of the Cartesian plane.
Coordinate Notation

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.
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.


A reflection is a transformation that flips a figure on the coordinate plane across a given line without changing the shape or size of the figure.


A rotation is a transformation that turns a figure on the coordinate plane a certain number of degrees about a given point without changing the shape or size of the figure.


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


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

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Difficulty Level:

At Grade


Date Created:

Nov 30, 2012

Last Modified:

Jan 26, 2016
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