What is Translation in Math? – Definition, Examples, & Terms

Translation Definition

Translation is a term used in geometry to describe a function that moves an object a certain distance. The object is not altered in any other way. It is not rotated, reflected or re-sized.

The rust-colored item is the pre-image, and the blue item is the image.

In a translation, every point of the object must be moved in the same direction and for the same distance.

When you are performing a translation, the initial object is called the pre-image, and the object after the translation is called the image. So, in the picture above, the rust-colored item is the pre-image, and the blue item is the image. We know this because the arrow tells us the direction in which the image was moved. For other images, you might be told which image is the pre-image, or you might be asked to find either the pre-image from the image, or vice versa.

How to Perform Translations

When working with translation problems, the information may be presented in different ways.

You may be given a figure drawn on the coordinate plane like this:

Image on coordinate plane

Then you will be asked to translate the figure. You will be given a distance and direction for the transformation.

For example, translate the figure down 7.

The way to do this is to take each vertex point individually and count down 7. So the point at (1, 5) will move to (1, -2). Notice we did not move the vertex along the x-axis, or horizontal direction. The instructions asked us to move it down only, along the y-axis.

Move the other three vertices in the same manner. The point at:

• (3, 5) moves to (3, -2)
• (1, 3) moves to (1, -4)
• (3, 3) moves to (3, -4)

Then connect the vertices to draw the square, translated down 7.

Image translated down 7

Another way that information might be given is like this, starting again with an image drawn on the coordinate plane:

Image on coordinate plane

This time, you will not be asked to draw the translation, but instead to describe it in mathematical notation.

Describe a translation of the triangle down 2 and to the right 3.

The notation will look like this:

(x, y) → (x + 3, y – 2)