8.19: Translations, Rotations, and Reflections
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.
Guidance
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
Let’s see why this happens.
We can see the change in all of the
We can translate figures in other ways, too. As you might guess, we move figures right or left on the coordinate grid by their
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
Now let’s compare some of the vertices. In the figure above the coordinates for the upperleft vertex of the original figure are (5, 5). After we reflect it across the
In a horizontal (left/right) reflection or a reflection over the
This is a reflection across the
We can recognize reflections by these changes to the
We can also use this information to graph reflections. To graph a reflection, we need to decide whether the reflection will be across the
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
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
Next, we need to count how many units long and wide the figure is. The figure above stretches from 1 on the
Now, remember the point (1, 3) stays the same, so it is one corner of the rotated figure. We add 5 to the
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
Write each set of coordinates to show a reflection in the
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
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
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.
Vocabulary
Here are the vocabulary words that are found in this Concept.
 Transformation
 a figure that is moved in the coordinate grid is called a transformation.
 Coordinate Plane

a representation of a twodimensional plane using an
x axis and ay axis.

x− axis  the horizontal line in a coordinate plane.

y− axis  the vertical line in a coordinate plane.
 Translation
 a slide. A figure is moved up, down, left or right.
 Reflection

a flip. A figure can be flipped over the
x− axis or they− axis.
 Rotation
 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
Answer
We need to reflect the rectangle across the
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.
Now we graph the new points.
Here is the completed reflection.
Practice
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
9. Rotate this figure
10. Reflect this figure over the
11. Reflect this figure over the
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
axis
The axis is the horizontal axis in the coordinate plane, commonly representing the value of the input or independent variable.axis
The axis is the vertical number line of the Cartesian plane.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
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.Reflection
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.Rotation
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.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.Image Attributions
Description
Learning Objectives
Here you'll learn to translate, rotate and reflect figures using coordinate notation and graphing.