A **transformation** of a figure is a **mapping** that changes the figure's size or position in space. We can also think of a transformation as a function, where the input values are the points that make up the figure that is being transformed.

The figure before it is transformed is called the **preimage**. The figure after it has been transformed is called the **image**. The points that make up the image would be considered the outputs of the function transformation.

Drag the points on the preimage to change its shape. The image is created by translating the preimage. Drag the sliders to translate the image horizontally and vertically.

If you are given the coordinates of the image, what information would you need to find the coordinates of the preimage and vice versa?

What can we conclude about the corresponding segment between any two points on the preimage and its image?

A **rigid transformation** (or **isometry**) is a transformation that does not change the size or shape of a figure.

It is common to label the vertices of figures with letters and to use an apostrophe, called a prime, to label vertices of the transformed image. For example, if A was the preimage, then A' (spoken as "A-prime") is the image.

We can describe a translation algebraically using coordinates:

Coordinate notation: The translation \left(x,y\right) \to \left(x+h,y+k\right) takes the preimage and moves it h units horizontally, and k units vertically to obtain the image.

The movement h units horizontal and k units vertical can be represented as a **directed line segment**. So we can think of a translation as moving the preimage along the directed line segment to get the image.

Since every point in the preimage is moved in the same direction and distance, every line segment from a preimage point to its corresponding image point will be parallel to the directed line segment that represents the translation.

For the following graph:

Describe the transformation in words.

Worked Solution

Triangle ABC is to be translated 4 units right and 2 units down.

a

Write the transformation in coordinate form.

Worked Solution

b

Draw the image.

Worked Solution

Draw the image given from the transformation \left(x,y\right) \to \left(x+1,\,y-4\right) on the preimage:

Worked Solution

A triangle is translated 5 units to the right and 3 units down. The coordinates of the image after this translation are S'\left(7, 2\right),\, T'\left(4, 0\right) and U'\left(2, 3\right).

Determine the coordinates of the preimage points S,T, and U before the translation.

Worked Solution

Idea summary

A translation is a rigid transformation in which an image is formed by moving every point on the preimage the same distance in the same direction.

Using coordinate notation: the translation \left(x,y\right) \to \left(x+h,\,y+k\right) takes the preimage and moves it h units horizontally, and k units vertically to obtain the image.