Leap Frog

Learning Focus

Compose transformations and describe a single motion that has the same result.

Big idea

A composition applies one transformation, then another to the image.

• Two translations compose to a translation.
• Reflections over two parallel lines compose to a translation of twice the distance between the lines.
• Reflections over two perpendicular lines compose to a $180^\circ$ rotation about their intersection.
• Order can matter, so always apply the motions in the stated order.

Worked example

Reflect $(a,b)$ over $x=0$, then over $x=4$.

• First: $(-a,b)$.
• Second: reflecting over $x=4$ sends $x\to 8-x$, giving $(8+a,\,b)$.

The result is a translation $8$ units right, twice the distance between the lines.

Takeaways

• Apply transformations in order.
• Parallel mirrors give a translation.
• Perpendicular mirrors give a half-turn.