Compositions of Transformations

Unit 9: Transformations · ~15–30 min

Combining transformations

A composition of transformations applies one transformation, then applies a second transformation to the result. Notation like (T \(\circ\) R)(P) means "apply R first, then apply T to that result" — work from the inside out, just like function composition in algebra. Order matters: reflecting then translating usually gives a different image than translating then reflecting.

A useful shortcut: reflecting twice

Two reflections over parallel lines are equivalent to a single translation perpendicular to those lines, with a distance equal to twice the distance between the lines. Two reflections over intersecting lines are equivalent to a single rotation about the point of intersection, through an angle twice the angle between the lines. These shortcuts explain why compositions of isometries are always themselves isometries — a composition of translations, reflections, and rotations can always be described as a single translation, rotation, reflection, or glide reflection.

Glide reflections

A glide reflection is the composition of a translation and a reflection over a line parallel to the translation vector. It's the transformation behind a row of footprints: each step is a reflection of the last, shifted forward.

Guided practice

• 1.Point C(−1, 4). Reflect over the x-axis, then translate by (x, y) → (x − 2, y + 1). Find the final image.
  Show answer

  Reflect: (−1, 4) → (−1, −4). Translate: (−1 − 2, −4 + 1) = (−3, −3).
• 2.Point D(5, 0). Rotate 90° counterclockwise about the origin, then reflect over the y-axis. Find the final image.
  Show answer

  Rotate: (5, 0) → (0, 5). Reflect over y-axis: (0, 5) → (0, 5) (unchanged, since x = 0).
• 3.Does composing a translation with a reflection always give the same result regardless of order? Explain.
  Show answer

  Not in general. Compositions of transformations usually depend on order, since each transformation acts on the output of the previous one; the reflection or translation used changes which points are being moved.
• 4.A figure is reflected over two parallel vertical lines, 3 units apart. What single transformation is this equivalent to?
  Show answer

  Two reflections over parallel lines equal a single translation perpendicular to the lines, with distance twice the gap between them: a horizontal translation of 6 units.
• 5.What two transformations make up a glide reflection?
  Show answer

  A translation and a reflection over a line parallel to the direction of the translation.