Composition of Transformations

Pre-Algebra · Axiom Academy

LESSON Composition of Transformations Apply one transformation, then feed its image into the next — and watch why the order you choose can change where you land. Take the point P=(1,2) and apply two transformations in sequence. First a translation T that slides every point right 3 and up 1. Then a reflection R across the x -axis, which flips the sign of the y -coordinate. The image after T is what R acts on. Move 1 — translate (slide right 3, up 1) Move 2 — reflect across the x -axis 2. Order Can Change the Result Unlike adding numbers, where 3+5=5+3 , swapping the order of two transformations usually lands you somewhere new. Watch the same point P=(1,2) travel down two routes — translate-then-reflect on the left, reflect-then-translate on the right. T(1,2)=(4,3) , then R(4,3)=(4,-3) . Lands at (4,-3) . R(1,2)=(1,-2) , then T(1,-2)=(4,-1) . Lands at (4,-1) . The reflection flips y to -y . If you slide before flipping, the +1 vertical slide gets flipped along with the point; if you slide after , it doesn't. That one difference is what separates the two landing spots. Order isn't always a trap. Some pairs of transformations commute — you get the same result either way. Two translations are the classic example: sliding right 3 / up 1 and then right 2 / up 2 reaches the same point as doing those slides in the opposite order.

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