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, where the output of one transformation becomes the input of the next. The order matters because transformation composition is generally not commutative.

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

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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: Composing transformations applies them in sequence, and the order usually changes the outcome.

Common stuck point: The procedure for composition of transformations is the easy part; the trap is applying transformations in the wrong order. Asking "Are two or more transformations applied one after another, with the output of one feeding the next?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are two or more transformations applied one after another, with the output of one feeding the next?

Worked Examples

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.

Answer

P′′=(1,3)P'' = (1, 3)

First step

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).

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Example 2

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

Example 3

medium
Point C(3,4)C(3,4) is reflected over the xx-axis, then over the yy-axis. Which single transformation gives the same image?

Example 4

medium
Show that reflecting over y=xy=x then over the xx-axis is NOT the same as doing them in the opposite order, using the point (2,5)(2,5).

Example 5

medium
A glide reflection on the xx-axis means: reflect over the xx-axis, then translate by ⟨a,0⟩\langle a,0\rangle. Apply with a=3a=3 to F(2,5)F(2,5).

Example 6

medium
Find a single transformation equal to reflecting over the yy-axis followed by reflecting over the line x=3x=3.

Example 7

hard
Find the image of (4,1)(4,1) under reflection in the line y=xy=x followed by translation ⟨−2,3⟩\langle -2,3\rangle, then explain why swapping the order generally fails.

Example 8

hard
Show that translation by ⟨6,0⟩\langle 6,0\rangle can be written as the composition of two reflections, and identify the two mirror lines.

Example 9

hard
A point is dilated by factor 33 about (0,0)(0,0), then dilated by factor 12\tfrac{1}{2} about (0,0)(0,0). What single dilation does this equal, and what is the image of (4,−2)(4,-2)?

Example 10

challenge
Prove that the composition of any rotation by θ\theta about AA followed by a rotation by −θ-\theta about BB (A≠BA\neq B, θ≠0\theta\neq 0) is a translation, and find the translation vector when A=(0,0)A=(0,0), B=(1,0)B=(1,0), θ=90°\theta=90°.

Practice Problems

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

Example 1

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

Example 2

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

Example 3

easy
Point A(4,−2)A(4,-2) is reflected over the yy-axis, then translated by ⟨0,5⟩\langle 0,5\rangle. Find A′′A''.

Example 4

easy
Point P(2,5)P(2,5) undergoes the translation ⟨−3,1⟩\langle -3,1\rangle followed by ⟨4,−2⟩\langle 4,-2\rangle. Find P′′P''.

Example 5

easy
Point B(−1,4)B(-1,4) is rotated 180°180° about the origin, then reflected over the xx-axis. Find B′′B''.

Example 6

medium
Apply to D(5,2)D(5,2): rotate 90°90° counterclockwise about the origin, then translate by ⟨3,−1⟩\langle 3,-1\rangle. Find D′′D''.

Example 7

medium
Point E(−3,2)E(-3,2) is reflected over the line y=xy=x, then rotated 90°90° clockwise about the origin. Find E′′E''.

Example 8

medium
A figure is dilated by scale factor 22 centered at the origin, then translated by ⟨4,−1⟩\langle 4,-1\rangle. Find the image of (3,5)(3,5).

Example 9

medium
Triangle with vertices A(1,1)A(1,1), B(4,1)B(4,1), C(1,3)C(1,3) is translated by ⟨2,3⟩\langle 2,3\rangle then reflected over the xx-axis. Find the new vertices.

Example 10

medium
Apply to (2,−3)(2,-3): rotate 90°90° counterclockwise about origin, then rotate 180°180° about origin. What single rotation does this equal, and what is the image?

Example 11

hard
Triangle XYZXYZ with X(0,0)X(0,0), Y(2,0)Y(2,0), Z(0,3)Z(0,3) is reflected over the yy-axis, then rotated 90°90° CCW about the origin. Find the image vertices.

Example 12

hard
Apply to (1,2)(1,2): rotate 90°90° CCW about (0,0)(0,0), then rotate 90°90° CCW about (1,0)(1,0). Find the image.

Example 13

medium
Apply to (6,−3)(6,-3): translation by ⟨−4,2⟩\langle -4,2\rangle, then rotation 180°180° about origin. Find the image.

Example 14

hard
Triangle ABCABC with A(1,2)A(1,2), B(3,2)B(3,2), C(2,5)C(2,5) is dilated by factor 22 about origin, then reflected over the xx-axis. Find the image vertices.
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Background Knowledge

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

translationrotationreflection