Enter Reflection Inputs
Example Data Table
| Original Point | Reflection | Rule Used | Reflected Point |
|---|---|---|---|
| (4, 3) | x-axis | (x, y) → (x, -y) | (4, -3) |
| (4, 3) | y-axis | (x, y) → (-x, y) | (-4, 3) |
| (4, 3) | origin | (x, y) → (-x, -y) | (-4, -3) |
| (4, 3) | line y = x | (x, y) → (y, x) | (3, 4) |
| (4, 3) | line x = 2 | (x, y) → (2k - x, y) | (0, 3) |
| (4, 3) | line y = 1 | (x, y) → (x, 2k - y) | (4, -1) |
Formula Used
Reflection flips a point across a mirror line while keeping the mirror line exactly halfway between the original and reflected points.
- Across x-axis: (x, y) → (x, -y)
- Across y-axis: (x, y) → (-x, y)
- Across origin: (x, y) → (-x, -y)
- Across y = x: (x, y) → (y, x)
- Across y = -x: (x, y) → (-y, -x)
- Across x = k: (x, y) → (2k - x, y)
- Across y = k: (x, y) → (x, 2k - y)
Distance moved is calculated with √[(x′ − x)² + (y′ − y)²]. The midpoint is ((x + x′)/2, (y + y′)/2).
How to Use This Calculator
- Enter the original x-coordinate and y-coordinate.
- Select the reflection type you want to apply.
- For custom mirror lines, enter the constant k.
- Set a graph range large enough to show both points.
- Press Reflect Graph to compute the transformation.
- Review the reflected coordinates, movement, midpoint, and graph.
- Use the CSV button for spreadsheet export.
- Use the PDF button for a printable result summary.
Frequently Asked Questions
1. What does reflecting a graph point mean?
It means creating a mirror image of a point across a chosen axis or line. The reflected point stays the same perpendicular distance from the mirror.
2. Why does reflection across the x-axis change only y?
The x-axis is horizontal, so the horizontal position stays fixed. Only the vertical distance reverses direction, making y become negative or positive opposite.
3. What happens when a point reflects across the y-axis?
The y-coordinate stays the same, while the x-coordinate changes sign. This creates a horizontal mirror image on the opposite side of the vertical axis.
4. How is reflection across y = x different?
Reflection across y = x swaps the coordinates. A point (a, b) becomes (b, a), which is useful for symmetry and transformation analysis.
5. What does the midpoint tell me?
The midpoint lies exactly on the mirror line for true reflections. It helps confirm the transformation and gives a geometric check for accuracy.
6. Why would I use a custom line x = k or y = k?
Custom lines help analyze symmetry away from the axes. They are useful in coordinate geometry exercises, classroom demonstrations, and engineering sketches.
7. What does the distance moved represent?
It measures how far the point traveled from its original position to the reflected position. Larger distances indicate the point was farther from the mirror.
8. Can this calculator help with teaching graph symmetry?
Yes. It combines formulas, coordinates, movement values, and a visual graph, making reflection concepts easier to explain and verify during lessons.