Reflection Formula

The Formula

Over x-axis: (x, y) \to (x, -y) Over y-axis: (x, y) \to (-x, y) Over y = x: (x, y) \to (y, x)

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

Quick Example

Reflect point (3, 2) over the y-axis: (3, 2) \to (-3, 2)—only the x-sign flips.

Notation

r_\ell denotes reflection over line \ell

What This Formula Means

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.

Formal View

Reflection over a line \ell through origin with unit direction \hat{u}: r_\ell(\vec{v}) = 2(\vec{v} \cdot \hat{u})\hat{u} - \vec{v}; over x-axis: r(x,y) = (x, -y); over y-axis: r(x,y) = (-x, y); \det(r_\ell) = -1

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.

Common Mistakes

  • Reflecting over the wrong axis — reflecting over the x-axis changes the y-coordinate, not the x-coordinate
  • Changing the distance to the mirror line — reflected points must be the same distance from the line as the originals
  • Forgetting that reflection reverses orientation — a reflected shape is a mirror image, not identical

Why This Formula Matters

Creates mirror images; forms the basis for symmetry analysis and transformation composition.

Frequently Asked Questions

What is the Reflection formula?

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

How do you use the Reflection formula?

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

What do the symbols mean in the Reflection formula?

r_\ell denotes reflection over line \ell

Why is the Reflection formula important in Math?

Creates mirror images; forms the basis for symmetry analysis and transformation composition.

What do students get wrong about Reflection?

Reflecting over the x-axis negates y; over the y-axis negates x; over y = x swaps coordinates.

What should I learn before the Reflection formula?

Before studying the Reflection formula, you should understand: transformation geo.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Symmetry, Rotational Symmetry, and Congruence →
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Related Concepts

Geometric Transformation