Reflection Calculator (Coordinate)
Enter a point and choose what to reflect it over to get the image point and the rule used.
Reflecting a point produces a mirror image across a line. Over the x-axis, (x, y) becomes (x, −y); over the y-axis, (−x, y); over the line y = x, (y, x); over y = −x, (−y, −x); and through the origin, (−x, −y). The point keeps its distance from the line, on the opposite side.
How to Use This Calculator
Enter the coordinates of your point and choose the line or point to reflect it over: the x-axis, the y-axis, the diagonal line y = x, the line y = −x, or the origin. The calculator returns the reflected point — called the image — and states the rule it applied.
Reflections are one of the basic rigid transformations in geometry: they flip a figure across a line without changing its size or shape.
How the Calculation Works
Each reflection has a simple coordinate rule. Reflecting over the x-axis flips the sign of y, because the point moves to the same horizontal position but the opposite vertical side. Over the y-axis it flips the sign of x.
Reflecting over y = x swaps the x and y coordinates, and over y = −x swaps and negates them. Reflecting through the origin negates both, which is the same as a 180-degree rotation. In every case the image is the same distance from the mirror as the original, on the other side.
x-axis (x,−y) · y-axis (−x,y) · y=x (y,x) · y=−x (−y,−x) · origin (−x,−y)
Worked Example
Reflect the point (3, 4) over the x-axis. The rule (x, y) → (x, −y) keeps the x the same and flips the y: (3, −4).
Reflecting the same point over y = x swaps the coordinates to (4, 3), and through the origin gives (−3, −4).
Assumptions and Limitations
- Reflects a single point in the 2D coordinate plane
- Supports the five most common mirrors (axes, y = ±x, origin)
- Reflection preserves distance and shape — it only flips position
- For an arbitrary line, a more general reflection formula is needed
Who Uses a Reflection Calculator
Geometry students use this to reflect a point over an axis or line and see the rule applied. It is built for transformation homework and for understanding how coordinates change under a flip. Enter a point and the line of reflection to get the image point and the mapping rule.
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Open calculator →Frequently Asked Questions
How do you reflect a point over the x-axis?
Keep the x-coordinate and flip the sign of the y-coordinate: (x, y) becomes (x, −y). So (3, 4) reflected over the x-axis is (3, −4).
What is the rule for reflecting over y = x?
Swap the coordinates: (x, y) becomes (y, x). Reflecting (3, 4) over the line y = x gives (4, 3). Over y = −x, you swap and negate to get (−y, −x).
Is reflecting over the origin the same as a rotation?
Yes. Reflecting a point through the origin, (x, y) → (−x, −y), gives the same result as rotating it 180 degrees about the origin.
Does reflection change a shape's size?
No. Reflection is a rigid transformation — it flips a figure across a line but keeps every distance and angle the same, so the image is congruent to the original.
How do I reflect a whole shape?
Apply the same reflection rule to each of its vertices, then connect the image points. This tool reflects one point at a time, which you repeat for each corner.
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