Combining Transformations

You can apply transformations one after another: take a shape, transform it, then transform the image again. For example, reflect a shape and then translate the result.

Often the combined effect of two transformations can be described as a single transformation:

two reflections in parallel mirror lines together make a translation;
two reflections in intersecting mirror lines together make a rotation about the point where they cross.

Be careful: the order usually matters. Doing transformation A then B generally lands the shape somewhere different from B then A.

Combining transformations

You can apply transformations one after another to the image of the previous one.
A single equivalent transformation often describes the combined effect.
Two reflections in parallel lines → a translation.
Two reflections in intersecting lines → a rotation about their intersection.
The order generally changes the result.

Seeing it on a grid

Start with the object triangle, reflect it in the y-axis to get image 1, then translate image 1 to get image 2. Step through to watch one transformation follow the other.