Composition of Transformations Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

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

Solution

  1. 1
    Step 1 — Reflect over the xx-axis: (x,y)→(x,−y)(x, y) \to (x, -y), so P(3,1)→P′(3,−1)P(3,1) \to P'(3,-1).
  2. 2
    Step 2 — Translate by ⟨−2,4⟩\langle -2, 4 \rangle: add −2-2 to xx and 44 to yy: P′(3,−1)→P′′(3−2, −1+4)=P′′(1,3)P'(3,-1) \to P''(3-2,\,-1+4) = P''(1, 3).
  3. 3
    The final image is P′′(1,3)P''(1, 3).

Answer

P′′=(1,3)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.

About Composition of Transformations

Composition of transformations applies two or more transformations in sequence to a figure, where the output of one transformation becomes the input of the next. The order matters because transformation composition is generally not commutative.

Learn more about Composition of Transformations →

More Composition of Transformations Examples

Example 2 hard

Triangle [formula] with [formula], [formula], [formula] is rotated [formula] counterclockwise about

Example 3 easy

Point [formula] is translated by [formula] and then reflected over the [formula]-axis. Find the fina

Example 4 medium

Describe the single transformation equivalent to two successive reflections over parallel lines [for

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