1.7 Composing Transformations — Transformations

Standards: G-CO.A.5 · G-CO.B.6

Given two transformations and an order, you will carry the output of one into the next, compare reversed orders, and determine whether the full sequence preserves congruence.

Definitions and rules

Composition — one transformation after another

(g ∘ f)(P) means do f first, then g

Read it right to left. The one nearest the point goes first.

Why it has to be this: the inner rule has to produce a point before the outer rule has anything to act on.

Why the order matters

Take (2, 3). Translate by T⟨3, −1⟩ then rotate 90°, and you land on (−2, 5). Rotate first and then translate, and you land on (0, 1).

translate then rotate: (−2, 5) rotate then translate: ( 0, 1)

Different answers from the same two transformations. Order is part of the instruction, not a detail.

Worked examples

Example 1 — order changes the answer

Apply T⟨3, −1⟩ to (2, 3), then rotate 90° counter-clockwise. Then do it in the opposite order.

Suggested work

Givens

(2, 3) T⟨3, −1⟩ and R90

Tools

T⟨a, b⟩(x, y) = (x + a, y + b) R90(x, y) = (−y, x)

Primitives

a = 3, b = −1

Computation

translate first: (2, 3) → (5, 2) R90(5, 2) = (−2, 5) rotate first: R90(2, 3) = (−3, 2) (−3, 2) → (0, 1) (−2, 5) is not (0, 1)

Notice: the same two transformations, two different destinations.

Example 2 — two translations collapse into one

Apply T⟨3, 0⟩ to (2, 3), then apply T⟨0, −5⟩ to the result.

Suggested work

Givens

(2, 3) first T⟨3, 0⟩, then T⟨0, −5⟩

Tools

T⟨a, b⟩(x, y) = (x + a, y + b), applied twice

Primitives

first: a = 3, b = 0 second: a = 0, b = −5

Computation

(2, 3) → ((2)+(3), (3)+(0)) = (5, 3) (5, 3) → ((5)+(0), (3)+(−5)) = (5, −2) and T⟨3, −5⟩(2, 3) = (5, −2)

Notice: two translations always collapse into one — add the vectors.

Why they collapse but rotations and reflections may not

Two translations commute and combine, because each is a single displacement and displacements add.

A translation followed by a rotation does not commute — see Example 1 (order changes the answer). Order matters as soon as the two transformations are of different kinds.

Example 3 — a translation followed by one reflection

Translate P(1, 2) by T⟨2, 1⟩, then reflect the result over the x-axis.

Suggested work

Givens

P(1, 2) T⟨2, 1⟩, then reflect over x-axis

Tools

translation: (x, y) → (x + 2, y + 1) reflection over x-axis: (x, y) → (x, −y)

Primitives

The translation output becomes the reflection input.

Computation

P(1, 2) → P′(3, 3) P′(3, 3) → P′′(3, −3)

Notice: The intermediate image is data for the next rule, not a separate final answer.

Why order is part of the instruction

The reflection acts on the translated coordinates. Changing which coordinates enter the reflection can change the destination.

Common traps

The intermediate image becomes the next input

Applying both transformations to the original point creates two separate images, not a composition. In a composition, the output of one rule becomes the input of the next. The broken assumption is treating each rule as independent when the instruction links them.

Practice

Reflect P(2, 1) over the y-axis, then apply T⟨3, 0⟩.
Answer — tap to show
P′′(1, 1).
Suggested work — tap to show
Givens
P(2, 1) reflect over y-axis, then T⟨3, 0⟩
Tools
over y-axis: (x, y) → (−x, y) T⟨3, 0⟩(x, y) = (x + 3, y)
Primitives
Use the reflected point as the translation input.
Computation
P(2, 1) → P′(−2, 1) P′(−2, 1) → P′′(1, 1)
Reflect (3, 1) over the x-axis, then rotate 180° about the origin.
Answer — tap to show
(−3, 1)
Suggested work — tap to show
Givens
(3, 1) reflect over x-axis, then R180
Tools
over the x-axis: (x, y) → (x, −y) R180(x, y) = (−x, −y)
Primitives
x = 3 y = 1
Computation
reflect: (3, 1) → (3, −1) rotate: (3, −1) → (−3, 1)
Apply T⟨2, 4⟩ to (1, 1), then T⟨−5, 3⟩.
Answer — tap to show
(−2, 8)
Suggested work — tap to show
Givens
(1, 1) then T⟨2, 4⟩, then T⟨−5, 3⟩
Tools
two translations — add the vectors
Primitives
first: a = 2, b = 4 second: a = −5, b = 3
Computation
(1, 1) → (3, 5) (3, 5) → (−2, 8) or in one step: T⟨−3, 7⟩(1, 1) = (−2, 8) ✓
Start with P(1, 2). You must use T⟨2, −1⟩ and R90. Which order reaches (−1, 3)?
Answer — tap to show
Translate, then rotate. (1, 2) → (3, 1) → (−1, 3).
Suggested work — tap to show
Givens
P(1, 2) T⟨2, −1⟩ R90 target: (−1, 3)
Tools
T⟨2, −1⟩(x, y) = (x + 2, y + (−1)) R90(x, y) = (−y, x)
Primitives
Test both possible orders.
Computation
translate, then rotate: (1, 2) → (3, 1) → (−1, 3) ✓ rotate, then translate: (1, 2) → (−2, 1) → (0, 0) Only translation followed by rotation reaches the target.
A classmate is asked to translate (1, 2) by T⟨2, 0⟩ and then rotate 180°. They rotate first and get (1, −2). Find their mistake.
Answer — tap to show
They reversed the order. The correct image is (−3, −2).
Suggested work — tap to show
Givens
(1, 2) translate T⟨2, 0⟩ first, THEN R180 Classmate got (1, −2)
Tools
T⟨a, b⟩(x, y) = (x + a, y + b) R180(x, y) = (−x, −y)
Primitives
a = 2, b = 0
Computation
correct order: (1, 2) → (3, 2) R180(3, 2) = (−3, −2) their order: R180(1, 2) = (−1, −2) (−1, −2) → (1, −2) The mistake: order is part of the instruction.