Composition of Transformations

Geometry · Axiom Academy

LESSON Composition of Transformations Combining two or more transformations into one — and why the order you apply them changes where a figure lands. 1. Applying One Transformation After Another A composition is a sequence of steps: transform the figure once, then transform the result again. The first transformation's output is the second one's input. Below, a triangle is moved by T_1 , and that intermediate figure is then moved by T_2 to its final position. Watch it travel, then read the notation. Read right to left: T_1 is applied first, then T_2 . In most cases, swapping the order of two transformations changes the final result. Here the same two moves — a translation by (4, 0) and a reflection over the y -axis — are applied in both orders . Watch the two copies separate and land in different places. Slide right by 4, then flip across the y -axis. The triangle ends up on the left. Flip across the y -axis first , then slide right by 4. It lands somewhere else. Composition is generally not commutative . 3. Example 1 — Two Translations To evaluate a composition on a point, apply the transformations in order and carry the result forward. Apply then to P(2, 4) . The animation walks P along the first shift vector, then the second. In Example 1: (3, -2) + (-1, 5) = (2, 3) , so the combined transformation is — exactly the dashed shortcut arrow above. 4. Example 2 — Reflection then Rotation

This is the written version of the interactive lesson above. See the full Geometry course.