What is a reflection in geometry?

A reflection is a transformation that flips a point or figure across a line (the line of reflection), producing a mirror image. The reflected point is the same distance from the line as the original, on the opposite side.

How do you reflect a point across the x-axis?

To reflect (x, y) across the x-axis, negate the y-coordinate: the result is (x, −y). The x-coordinate stays the same.

How do you reflect a point across the y-axis?

To reflect (x, y) across the y-axis, negate the x-coordinate: the result is (−x, y). The y-coordinate stays the same.

How do you reflect a point across y=x?

Reflecting (x, y) across the line y=x swaps the two coordinates: the result is (y, x).

How is the reflection across a custom line y=mx+b calculated?

The formula projects the point onto the line, computes the foot of the perpendicular, then reflects through it. If the foot is (fx, fy), the reflected point is (2fx−x, 2fy−y).

Does the order of reflection matter for multiple reflections?

Yes. Composing two reflections is generally not commutative — the result depends on the order in which you apply them.

Reflection Calculator

Name: Reflection Calculator
Availability: InStock
Author: Nham Vu

Enter a point and choose a line of reflection to get the mirror image coordinates instantly.

Input

Point to reflect

Line of reflection

x-axis (y = 0) y-axis (x = 0) Line y = x Line y = −x Origin (0, 0) Custom line y = mx + b

Result

Enter a point and click Reflect

Step-by-Step

Click "Reflect Point" to see the work.

Diagram

Blue = original | Red = reflected | Dashed = line of reflection

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Summary

Enter a point and choose a line of reflection to get the mirror image coordinates instantly.

How it works

Enter the x and y coordinates of your point.
Choose a line of reflection from the preset options or select "Custom line y=mx+b".
For a custom line, enter the slope (m) and y-intercept (b).
Click "Reflect" to compute the mirror image point.
View the reflected coordinates and the step-by-step derivation.

Use cases

Verify reflections in geometry homework or exams.
Check symmetry of figures across coordinate axes.
Compute reflections for computer graphics transformations.
Explore the effect of different lines of reflection on a point.
Practice and understand the reflection formula y=mx+b.
Visualize mirrored points on the coordinate plane.

Frequently Asked Questions

Last updated: 2026-07-24 · Reviewed by Nham Vu