Compositions of Transformations
Perform compositions of two or more transformations and find their net effect. Understand that the composition of two reflections across parallel lines is a translation, and across intersecting lines is a rotation.
A lesson of Geometry, in Transformations, about 10 minutes.
By the end of this lesson, you will be able to perform compositions of two or more transformations, identify the single transformation equivalent to a given composition, and apply the glide reflection concept.
The Big Idea
Two transformations applied in sequence form a composition. Mathematically, compositions are denoted right-to-left (apply the rightmost transformation first). Every composition of rigid motions is itself a rigid motion — and every rigid motion is either a translation, rotation, reflection, or glide reflection.
Notation
$(T_2 \circ T_1)(P)$ means "apply $T_1$ to $P$ first, then apply $T_2$ to the result."
Key Composition Results
Two reflections over parallel lines = a translation
The translation vector is twice the distance between the lines, perpendicular to them.
Two reflections over intersecting lines = a rotation
The rotation center is the intersection point; the angle is twice the angle between the lines.
Glide reflection: a translation followed by a reflection over a line parallel to the translation direction (or vice versa). It is the fourth type of isometry.
Example 1 — Two Reflections Over Parallel Lines
Reflect $P(2, 5)$ first over $x = 0$ (the $y$-axis), then over $x = 3$.
Step 1: Over $y$-axis: $P \to P' = (-2, 5)$
Step 2: Over $x = 3$: $(2(3) - (-2), 5) = (8, 5) = P''$
Net result: $(2, 5) \to (8, 5)$ — a translation of $\langle 6, 0 \rangle$.
The lines $x = 0$ and $x = 3$ are 3 apart; the translation is $2 \times 3 = 6$ in the direction perpendicular to the lines (horizontal). ✓
Example 2 — Two Reflections Over Intersecting Lines
Reflect $P(4, 0)$ over the $x$-axis, then over the line $y = x$.
Step 1: Over $x$-axis: $P = (4, 0) \to P' = (4, 0)$ (point is on $x$-axis — maps to itself)
Wait — $P = (4, 0)$ is on the $x$-axis, so it's fixed. Let's use $P(3, 1)$.
Step 1: Over $x$-axis: $(3, 1) \to P' = (3, -1)$
Step 2: Over $y = x$: $(3, -1) \to P'' = (-1, 3)$
The $x$-axis and $y = x$ intersect at the origin at $45°$. The composition is a rotation of $2 \times 45° = 90°$ CCW about the origin.
Check: $90°$ CCW rule on $(3, 1)$: $(-1, 3)$ ✓
Example 3 — Glide Reflection
A glide reflection: translate by $\langle 4, 0 \rangle$ then reflect over the $x$-axis.
Apply to $P(1, 3)$:
Translation: $(1 + 4, 3) = (5, 3)$
Reflection over $x$-axis: $(5, -3)$