Reflection Formula

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

The Formula

across y-axis: (x,y)↦(−x,y)\text{across y-axis: }(x,y)\mapsto(-x,y)

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)(3, 2) over the yy-axis: (3,2)→(−3,2)(3, 2) \to (-3, 2)—only the xx-sign flips.

Notation

A reflection needs a mirror line; corresponding points are the same distance from that line.

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 ℓ\ell through origin with unit direction u^\hat{u}: rℓ(v⃗)=2(v⃗⋅u^)u^−v⃗r_\ell(\vec{v}) = 2(\vec{v} \cdot \hat{u})\hat{u} - \vec{v}; over xx-axis: r(x,y)=(x,−y)r(x,y) = (x, -y); over yy-axis: r(x,y)=(−x,y)r(x,y) = (-x, y); det⁡(rℓ)=−1\det(r_\ell) = -1

Worked Examples

Example 1

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

Answer

P′=(3,4)P' = (3, 4)

First step

1
Step 1: Reflecting over the x-axis: the rule is (x,y)→(x,−y)(x, y) \to (x, -y).

Full solution

  1. 2
    Step 2: Apply to P(3,−4)P(3, -4): P′=(3,−(−4))=(3,4)P' = (3, -(-4)) = (3, 4).
  2. 3
    Step 3: The point flips from below to above the x-axis.
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)Q(-2, 5) over the line y=xy = x. Find the image.

Example 3

medium
Reflect (2,7)(2, 7) across the horizontal line y=4y = 4.

Common Mistakes

  • Changing both coordinates for an axis reflection without checking the axis — across y changes x; across x changes y.
  • Reflecting across the wrong line — name the mirror line first.
  • Sliding instead of flipping — a reflection reverses orientation.

Why This Formula Matters

Reflections connect symmetry, coordinate rules, congruence, and geometric proof. Students learn to identify what changes and what stays invariant. Recognizing it by "Can I identify the mirror line and equal distances from it?" — rather than by familiar numbers — is what lets a student tell it apart from translation and rotation in a mixed problem set.

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?

A reflection needs a mirror line; corresponding points are the same distance from that line.

Why is the Reflection formula important in Math?

Reflections connect symmetry, coordinate rules, congruence, and geometric proof. Students learn to identify what changes and what stays invariant. Recognizing it by "Can I identify the mirror line and equal distances from it?" — rather than by familiar numbers — is what lets a student tell it apart from translation and rotation in a mixed problem set.

What do students get wrong about Reflection?

The procedure for reflection is the easy part; the trap is changing both coordinates for an axis reflection without checking the axis. Asking "Can I identify the mirror line and equal distances from it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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