Reflection Examples in Math

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

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

Concept Recap

A rigid transformation that flips a figure over a line (the mirror line), producing a mirror image.

Like looking in a mirror—left and right are swapped, but size and shape are perfectly preserved.

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: Reflection reverses orientation while preserving size and shape.

Common stuck point: Reflecting over the x-axis negates y; over the y-axis negates x; over y = x swaps coordinates.

Sense of Study hint: Draw the mirror line, then for each point measure the perpendicular distance to the line and plot the same distance on the other side.

Worked Examples

Example 1

easy
Reflect the point P(3, -4) over the x-axis. What are the coordinates of the image?

Solution

  1. 1
    Step 1: Reflecting over the x-axis: the rule is (x, y) \to (x, -y).
  2. 2
    Step 2: Apply to P(3, -4): P' = (3, -(-4)) = (3, 4).
  3. 3
    Step 3: The point flips from below to above the x-axis.

Answer

P' = (3, 4)
Reflecting over the x-axis keeps the x-coordinate the same and negates the y-coordinate. The x-axis acts as a mirror: points below the x-axis map to the corresponding points above it, and vice versa.

Example 2

medium
Reflect the point Q(-2, 5) over the line y = x. Find the image.

Practice Problems

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

Example 1

easy
Reflect (5, 3) over the y-axis. What is the image?

Example 2

hard
Reflect the point R(4, 1) over the line y = 2. Find the image coordinates and explain your method.
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Related Concepts

Geometric Transformation

Background Knowledge

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

transformation geo