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Parabola Calculator
Find every key feature of a parabola from its equation: vertex, focus, directrix, axis of symmetry, focal distance, focal width and intercepts, in exact fractions, with a graph. Works for vertical parabolas y = ax² + bx + c and horizontal ones x = ay² + by + c.
Last updated:
Equation form
Orientation
Whole numbers, decimals or fractions like 1/8.
Vertex (h, k). Horizontal form: x = a(y − k)² + h.
Enter your values to see the result.
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How to use the Parabola Calculator
- Choose the form: standard ax² + bx + c or vertex a(x − h)² + k.
- Choose the orientation: vertical (opens up or down) or horizontal (opens left or right).
- Enter the coefficients; fractions like 1/4 are allowed.
- Read the vertex, focus, directrix and other properties.
- Check the graph and the table of points.
Parabola formulas
| Property | Vertical: y = a(x − h)² + k | Horizontal: x = a(y − k)² + h |
|---|---|---|
| Vertex | (h, k) | (h, k) |
| From standard form | h = −b/(2a), k = c − b²/(4a) | k = −b/(2a), h = c − b²/(4a) |
| Focal distance | p = 1/(4a) | |
| Focus | (h, k + p) | (h + p, k) |
| Directrix | y = k − p | x = h − p |
| Axis of symmetry | x = h | y = k |
| Opens | up if a > 0, down if a < 0 | right if a > 0, left if a < 0 |
| Focal width | |1/a| = |4p| | |
| 4p form | (x − h)² = 4p(y − k) | (y − k)² = 4p(x − h) |
Examples
| Equation | Vertex | Focus | Directrix |
|---|---|---|---|
| y = x² | (0, 0) | (0, 1/4) | y = −1/4 |
| y = (1/4)x² − x − 3 | (2, −4) | (2, −3) | y = −5 |
| y = −2x² + 8x − 3 | (2, 5) | (2, 39/8) | y = 41/8 |
| x = y² − 4y + 1 | (−3, 2) | (−11/4, 2) | x = −13/4 |
Frequently asked questions
How do you find the focus of a parabola?
Write the parabola in vertex form y = a(x − h)² + k. The focal distance is p = 1/(4a), and the focus is (h, k + p). For y = x², a = 1, p = 1/4 and the focus is (0, 1/4).
What is the directrix?
The directrix is the line on the opposite side of the vertex from the focus, at the same distance p. For a vertical parabola it is y = k − p. Every point on a parabola is equally far from the focus and the directrix.
How do I find the vertex from y = ax² + bx + c?
The x-coordinate is h = −b/(2a); substitute it back or use k = c − b²/(4a) for the y-coordinate. For y = −2x² + 8x − 3, h = 2 and k = 5, so the vertex is (2, 5).
What is the latus rectum?
It is the chord through the focus parallel to the directrix. Its length, the focal width, is |1/a| = 4|p|, which tells you how wide the parabola is at the level of the focus.
How is a horizontal parabola different?
A horizontal parabola has the form x = a(y − k)² + h and opens to the right when a > 0 or to the left when a < 0. Its vertex is (h, k), its focus is (h + p, k) with p = 1/(4a), and its directrix is the vertical line x = h − p.
What is the 4p form of a parabola?
The form (x − h)² = 4p(y − k) is the same parabola as y = a(x − h)² + k with a = 1/(4p). It is common in geometry and physics, for example for satellite dishes and headlights, where the focus is p units from the vertex.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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