4.5 Compositions of Transformations

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Notes

A composition is one transformation followed by another. Do them in order, using each image as the next preimage.

Order often matters: a translation then a reflection may differ from the reverse.
A glide reflection is a translation followed by a reflection over a line parallel to the translation.
Reflecting over two parallel lines equals a translation. Reflecting over two intersecting lines equals a rotation (twice the angle between them).
Example: $(1, 2)$, translate $(x + 3, y)$ to $(4, 2)$, then reflect over the $x$-axis to $(4, -2)$.

$P(2, 3)$ is reflected over the $y$-axis, then translated by $(x, y) \to (x + 5, y - 1)$. Find the final image.

A figure is reflected over line $m$ and then over line $n$, where $m \parallel n$. The result is equivalent to a single:

$Q(1, -4)$ is rotated $180^\circ$ about the origin, then dilated by a scale factor of 2 centered at the origin. Find the final image.