Composition of Transformations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Composition of Transformations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Composition of transformations applies two or more transformations in sequence to a figure.

Order matters, like doing rotate then reflect versus reflect then rotate.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The output of one transformation becomes the input of the next.

Common stuck point: Transformation composition is not commutative—T \circ R and R \circ T generally give different results.

Sense of Study hint: Label points after each step to track order explicitly.

Worked Examples

Example 1

medium
Point P(3, 1) is first reflected over the x-axis, then translated by vector \langle -2, 4 \rangle. Find the final image.

Solution

  1. 1
    Step 1 — Reflect over the x-axis: (x, y) \to (x, -y), so P(3,1) \to P'(3,-1).
  2. 2
    Step 2 — Translate by \langle -2, 4 \rangle: add -2 to x and 4 to y: P'(3,-1) \to P''(3-2,\,-1+4) = P''(1, 3).
  3. 3
    The final image is P''(1, 3).

Answer

P'' = (1, 3)
In a composition of transformations, each transformation is applied in sequence to the result of the previous one. Order matters: reflecting first then translating gives a different result than translating first.

Example 2

hard
Triangle ABC with A(1,0), B(3,0), C(2,2) is rotated 90° counterclockwise about the origin, then reflected over the y-axis. Find the final vertices.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Point Q(2, -3) is translated by \langle 1, 5 \rangle and then reflected over the x-axis. Find the final image.

Example 2

medium
Describe the single transformation equivalent to two successive reflections over parallel lines x = 1 and x = 4.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

translationrotationreflection