What it does
A reflection flips a point or a shape over a line, like a mirror image. This calculator takes one point, or a list of points that make a polygon, and reflects them over the x-axis, the y-axis, the lines y = x and y = −x, a vertical or horizontal line, any slanted line, or through the origin. It gives exact coordinates (fractions stay fractions), the rule it used, and a graph of the shape before and after. It's useful for geometry homework on transformations and for checking a drawing.
The reflection rules
| Reflect over | Rule | (2, 3) becomes |
|---|---|---|
| x-axis | (x, y) → (x, −y) | (2, −3) |
| y-axis | (x, y) → (−x, y) | (−2, 3) |
| y = x | (x, y) → (y, x) | (3, 2) |
| y = −x | (x, y) → (−y, −x) | (−3, −2) |
| Origin | (x, y) → (−x, −y) | (−2, −3) |
| x = k | (x, y) → (2k − x, y) | k = 5: (8, 3) |
| y = k | (x, y) → (x, 2k − y) | k = 1: (2, −1) |
For any line written as ax + by + c = 0, the reflection moves the point straight across the line, along the direction (a, b) that's perpendicular to it, by twice its distance from the line:
x′ = x − 2a·d y′ = y − 2b·d
A line y = mx + b is the same as mx − y + b = 0, so it uses a = m, b = −1 and c = b.
Worked example
Reflect (3, 4) over the line x + y − 1 = 0, so a = 1, b = 1, c = −1. Then ax + by + c = 3 + 4 − 1 = 6 and a² + b² = 2, so d = 3. The image is x′ = 3 − 2 × 1 × 3 = −3 and y′ = 4 − 2 × 1 × 3 = −2: (−3, −2). Check: the midpoint of (3, 4) and (−3, −2) is (0, 1), which is on the line, and the segment between them has slope 1, at a right angle to the line's slope of −1.
Properties of a reflection
- The mirror line is the perpendicular bisector of the segment from each point to its image.
- Points on the mirror line stay where they are.
- Lengths, angles and areas don't change, but the shape is flipped: corners labeled A, B, C counterclockwise come out clockwise.
- Reflecting twice over the same line brings every point back to where it started.
Limits
- This is for 2D points on a coordinate plane. Up to 26 points can be entered, labeled A to Z.
- Reflection through the origin is the same as a 180° rotation, not a flip over a line; it's included because many courses list it with reflections.
- Graphs of functions are reflected point by point the same way: −f(x) flips over the x-axis, f(−x) over the y-axis.