Lesson 4.2.2

Compositions of Transformations

A composition applies two or more transformations in sequence. The output of the first transformation becomes the input of the second — order matters!

Introduction

So far, each lesson covered a single transformation. But what happens when you chain them? Translate, then reflect. Reflect, then rotate. The result is a composition — and understanding compositions is the gateway to proving congruence.

Past Knowledge

Translations (4.1.2), reflections (4.1.4–5), rotations (4.1.6–7).

Today's Goal

Apply two or more transformations in sequence and describe the resulting composition.

Future Success

Glide reflections (4.2.3), congruence proofs (4.2.4), and sequence identification (4.2.5).

Key Concepts

Composition Notation

Read right to left: apply first, then to the result. The circle means “composed with.”

Key Facts About Compositions

The composition of two rigid transformations is always rigid.
Order matters — reflecting then translating usually gives a different result than translating then reflecting.
Two reflections across parallel lines = a translation.
Two reflections across intersecting lines = a rotation (angle = 2× the angle between the lines).

Double Reflection Theorems

Parallel lines and with distance apart: = translation by perpendicular to the lines.

Intersecting lines forming angle : = rotation of about the intersection point.

Worked Examples

Basic

Translation Then Reflection

Apply the composition: first translate with , , by , then reflect across the x-axis.

Step 1 — Translate by :

Step 2 — Reflect across x-axis:

T⟨3,0⟩

→Reflect x-axis

A(1,2)→A′′(4,-2)

B(4,2)→B′′(7,-2)

C(2,5)→C′′(5,-5)

Intermediate

Double Reflection (Parallel Lines)

Reflect across the y-axis, then reflect the result across the line . Describe the single transformation equivalent.

Step 1 — Reflect across y-axis:

Step 2 — Reflect across :

The y-axis and intersect at the origin at a 45° angle. By the double reflection theorem, this equals a rotation of CW (or 270° CCW) about the origin.

Verify: ✓

. Equivalent to a 270° CCW rotation about the origin.

Reflect y-axis

→Reflect y=x

P(2,3)→P′′(3,-2)

Advanced

Three-Step Composition

Apply to with , , : (1) reflect across y-axis, (2) rotate 90° CCW about origin, (3) translate by .

Step 1 — Reflect y-axis:

Step 2 — Rotate 90° CCW:

Step 3 — Translate :

Reflect y-axis

→Rot 90°

→T⟨2,-1⟩

D(1,1)→D′′′(1,-2)

E(3,1)→E′′′(1,-4)

F(2,4)→F′′′(-2,-3)

Common Pitfalls

Applying Transformations in the Wrong Order

Composition notation means apply first, then . Read right to left — but when given “first… then…” in words, follow that order.

Assuming Order Doesn't Matter

Translating then reflecting generally gives a different result than reflecting then translating. Always apply transformations in the specified sequence.

Losing Track of Intermediate Points

Label each intermediate result clearly ( after step 1, after step 2). Working neatly prevents sign errors from compounding.

Real-Life Applications

3D Animation Pipelines

Every 3D character in a movie or game has transformations composed in a specific order: scale the model, rotate the joints, then translate to the scene position. Changing the order produces completely different (and often hilarious) results.

Kaleidoscopes

A kaleidoscope works by composing reflections across two or three mirrors angled toward each other. The double reflection theorem explains why you see rotational patterns — the reflections compose into rotations.