# Where do points go in a reflection?

Reflects points in the coordinate plane across the x-axis, the y-axis, the origin, y = x, y = −x, a line x = h or y = k, or any line y = mx + b, with the rule and the working.

- Page: https://www.acalculator.org/math/reflection-calculator
- JSON spec: https://www.acalculator.org/math/reflection-calculator.json
- Version: 937d2fce8d07

## Default answer

Example with the default inputs (Reflect across x-axis, Points [x 3, y 2; x 5, y 2; x 4, y 6]): The image points are A′(3, −2); B′(5, −2); C′(4, −6).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| line | Reflect across | The line (or point) of reflection. |
| h | h | Where the vertical line x = h crosses the x-axis. |
| k | k | Where the horizontal line y = k crosses the y-axis. |
| m | Slope m | The slope of the line y = mx + b. |
| b | Intercept b | Where the line y = mx + b crosses the y-axis. |
| points | Points | The points of the figure to reflect, A, B, C and so on, up to 12. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| image | Image points | Each point after the reflection, as A′(x, y); B′(x, y) and so on. |
| rule | Rule | Where a point (x, y) goes, for the chosen line. |
| steps | Working | Each point put through the rule. |

## Method

x-axis (x, −y); y-axis (−x, y); origin (−x, −y); y = x (y, x); y = −x (−y, −x); x = h (2h − x, y); y = k (x, 2k − y); y = mx + b: d = (x + m(y − b)) ÷ (1 + m²), (2d − x, 2md − y + 2b).

## Assumptions

- Typed decimals are read exactly, so each image coordinate is the exact value, rounded once to 10 significant figures.
- A reflection keeps every length and angle; it flips the figure over the line.

## Worked examples

1. line = x, points = {"x":3,"y":2} or undefined gives image = A′(3, −2), rule = (x, y) → (x, −y). Source: CK-12 Foundation, Geometry, 8.14 Rules for Reflections ((x, y) → (x, −y) over the x-axis, (−x, y) over the y-axis, (y, x) over y = x, (−y, −x) over y = −x; (3, 2) over the x-axis is (3, −2), over y = x is (2, 3)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/08%3A_Rigid_Transformations/8.14%3A_Rules_for_Reflections (retrieved 2026-10-05).
2. line = yx, points = {"x":3,"y":2} or undefined gives image = A′(2, 3). Source: CK-12 Foundation, Geometry, 8.14 Rules for Reflections ((x, y) → (x, −y) over the x-axis, (−x, y) over the y-axis, (y, x) over y = x, (−y, −x) over y = −x; (3, 2) over the x-axis is (3, −2), over y = x is (2, 3)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/08%3A_Rigid_Transformations/8.14%3A_Rules_for_Reflections (retrieved 2026-10-05).
3. line = y, points = {"x":3,"y":2} or {"x":-1.5,"y":4} gives image = A′(−3, 2); B′(1.5, 4). Source: OpenStax, Algebra and Trigonometry 2e, §3.5 Transformation of Functions (−f(x) reflects about the x-axis, f(−x) about the y-axis), https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-5-transformation-of-functions (retrieved 2026-10-05).
4. line = v, h = 2, points = {"x":5,"y":-1} or undefined gives image = A′(−1, −1).
5. line = m, m = 2, b = 1, points = {"x":3,"y":2} or undefined gives image = A′(−1, 4).

## FAQ

### How do I reflect a point over the x-axis?

Keep x and change the sign of y: (x, y) → (x, −y). The point (3, 2) reflected over the x-axis is (3, −2).

### How do I reflect a point over the y-axis?

Change the sign of x and keep y: (x, y) → (−x, y). The point (3, 2) goes to (−3, 2).

### What is the rule for a reflection over y = x?

Swap the coordinates: (x, y) → (y, x). The point (3, 2) goes to (2, 3). Over y = −x, swap them and change both signs: (x, y) → (−y, −x).

### How do I reflect over a line such as x = 2 or y = −1?

Over the vertical line x = h, the image is (2h − x, y); over the horizontal line y = k, it is (x, 2k − y). The point (5, −1) reflected over x = 2 is (4 − 5, −1) = (−1, −1).

### How do I reflect a point over any line y = mx + b?

Find d = (x + m(y − b)) ÷ (1 + m²), the x coordinate of the point on the line closest to (x, y). The image is (2d − x, 2md − y + 2b). For (3, 2) and y = 2x + 1, d = 1 and the image is (−1, 4).

### Is a reflection over the origin a reflection over a line?

No. Reflecting through the origin sends (x, y) to (−x, −y), the same as a 180° rotation about the origin. It is listed because textbooks call it a point reflection.

## Sources

- CK-12 Foundation, Geometry, 8.14 Rules for Reflections (the rules over the x-axis, the y-axis, y = x and y = −x; (3, 2) over the x-axis is (3, −2) and over y = x is (2, 3)), on K12 LibreTexts. https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/08%3A_Rigid_Transformations/8.14%3A_Rules_for_Reflections (retrieved 2026-10-05)
- OpenStax, Algebra and Trigonometry 2e, §3.5 Transformation of Functions (−f(x) is a reflection about the x-axis and f(−x) about the y-axis), CC BY 4.0. https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-5-transformation-of-functions (retrieved 2026-10-05)