The Shape Reflection Calculator calculates the mirror image of a shape’s points across a line of reflection.
Each point is flipped using the rule for the chosen line — for example (x, y) → (x, −y) over the x-axis, or (x, y) → (2k − x, y) over the vertical line x = k. Choose the x-axis, y-axis, y = x, y = −x, the origin, or any line x = k or y = k, and the calculator lists the image points and graphs both shapes.
Shape Reflection Calculator
Reflect a shape over the x-axis, y-axis, y = x, y = −x, the origin, or any vertical or horizontal line, with a graph of the image.
Formula
| Reflection over | Rule |
|---|---|
| x-axis | (x, y) → (x, −y) |
| y-axis | (x, y) → (−x, y) |
| Line y = x | (x, y) → (y, x) |
| Line y = −x | (x, y) → (−y, −x) |
| Origin (point reflection) | (x, y) → (−x, −y) |
| Vertical line x = k | (x, y) → (2k − x, y) |
| Horizontal line y = k | (x, y) → (x, 2k − y) |
Where (x, y) is the original point and k is where the vertical or horizontal mirror line crosses the axis.
How the Calculation Works
Step 1: Measure each point’s distance to the mirror line.
Step 2: Place the image the same distance on the opposite side, along a path perpendicular to the line.
Step 3: Repeat for every point and join them in the same order.
Variables Explained
| Input | Meaning |
|---|---|
| Points | One x, y pair per line |
| Line of reflection | The mirror line |
| k | Position of the line x = k or y = k |
Example Calculation
Over the x-axis
Given: A(1, 1), B(5, 1), C(2, 4)
Result: A′(1, −1), B′(5, −1), C′(2, −4)
Over the line y = x
Result: A′(1, 1), B′(1, 5), C′(4, 2) — A lies on the line, so it stays put
Over the line x = 3
B(5, 1) → (2 × 3 − 5, 1) = B′(1, 1)
Shape Properties
A reflection is a rigid transformation: the image has the same size and shape, but its orientation is reversed (clockwise labels become counterclockwise). Points on the mirror line don’t move.
Dimensions Explained
Reflection over the origin
Reflecting through the origin gives the same result as a 180° rotation about the origin.
Two reflections
Reflecting over the x-axis and then the y-axis also gives (−x, −y). Reflecting over two parallel lines gives a translation.
Frequently Asked Questions
What is the rule for reflecting over the x-axis?
Keep x and change the sign of y: (x, y) → (x, −y).
What is the rule for reflecting over the y-axis?
(x, y) → (−x, y).
How do you reflect over y = x?
Swap the coordinates: (x, y) → (y, x).
How do you reflect over a line like x = 2?
Use (x, y) → (2 × 2 − x, y) = (4 − x, y).
Does a reflection change the size of a shape?
No. Only its position and orientation change.