# What is the focus of my parabola?

Finds a parabola’s vertex, focus, directrix, axis of symmetry and latus rectum from y = ax² + bx + c, x = ay² + by + c, or the standard form (x − h)² = 4p(y − k), with both equations and a graph.

- Page: https://www.acalculator.org/math/parabola-calculator
- JSON spec: https://www.acalculator.org/math/parabola-calculator.json
- Version: f15d40d48284

## Default answer

Example with the default inputs (Opens Up or down, I know a, b, c, a 2, b -6, c 7): The parabola y = 2x² − 6x + 7 has its focus at (3/2, 21/8) and its directrix at y = 19/8.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| axis | Opens | Up or down (y in terms of x), or left or right (x in terms of y). |
| form | I know | The coefficients a, b, c, or the vertex (h, k) and p. |
| a | a | The coefficient of the squared term. It may not be 0. |
| b | b | The coefficient of the first-power term. |
| c | c | The constant term. |
| h | h | The x-coordinate of the vertex. |
| k | k | The y-coordinate of the vertex. |
| p | p | The signed distance from the vertex to the focus. It may not be 0. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| focus | Focus | The focus, as exact fractions. |
| vertex | Vertex | The turning point (h, k). |
| directrix | Directrix | The line the parabola keeps the same distance from as from the focus. |
| axisLine | Axis of symmetry | The line through the vertex and the focus. |
| opens | Opens | Which way the parabola opens. |
| pValue | p | The signed distance from the vertex to the focus: 1 ÷ (4a). |
| latus | Latus rectum length | \|4p\|: the width of the parabola at the focus. |
| ends | Latus rectum endpoints | Where the chord through the focus, parallel to the directrix, meets the parabola. |
| standard | Equation in a, b, c | y = ax² + bx + c, or x = ay² + by + c. |
| vertexEq | Standard form | (x − h)² = 4p(y − k), or (y − k)² = 4p(x − h). |
| hx | h (vertex x) | The x-coordinate of the vertex, as a decimal. |
| ky | k (vertex y) | The y-coordinate of the vertex, as a decimal. |

## Method

Up or down: (x − h)² = 4p(y − k), focus (h, k + p), directrix y = k − p. Left or right: (y − k)² = 4p(x − h), focus (h + p, k), directrix x = h − p. From a, b, c: p = 1 ÷ (4a), vertex along the axis −b ÷ (2a), across it c − b² ÷ (4a).

## Assumptions

- The axis of the parabola is vertical or horizontal (no rotated parabolas).
- Coefficients are read as the exact decimals typed, so every answer is an exact fraction.

## Worked examples

1. axis = vertical, form = vertex, h = 4, k = -8, p = 7 gives vertex = (4, −8), focus = (4, −1), directrix = y = −15, axisLine = x = 4, ends = (−10, −1) and (18, −1), vertexEq = (x − 4)² = 28(y + 8), standard = y = (1/28)x² − (2/7)x − 52/7. Source: OpenStax, Algebra and Trigonometry 2e, §12.3 The Parabola (standard forms (x − h)² = 4p(y − k) with focus (h, k + p) and directrix y = k − p, and (y − k)² = 4p(x − h) with focus (h + p, k) and directrix x = h − p; latus rectum endpoints (h ± 2p, k + p) and (h + p, k ± 2p)), https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-3-the-parabola (retrieved 2026-10-02) (Example 5: vertex (4, −8), focus (4, −1), directrix y = −15, endpoints (−10, −1) and (18, −1)).
2. axis = horizontal, form = vertex, h = -3, k = 1, p = -4 gives focus = (−7, 1), directrix = x = 1, axisLine = y = 1, opens = Left, ends = (−7, −7) and (−7, 9), vertexEq = (y − 1)² = −16(x + 3), latus = 16. Source: OpenStax, Algebra and Trigonometry 2e, §12.3 The Parabola (standard forms (x − h)² = 4p(y − k) with focus (h, k + p) and directrix y = k − p, and (y − k)² = 4p(x − h) with focus (h + p, k) and directrix x = h − p; latus rectum endpoints (h ± 2p, k + p) and (h + p, k ± 2p)), https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-3-the-parabola (retrieved 2026-10-02) (Example 4: vertex (−3, 1), focus (−7, 1), directrix x = 1, endpoints (−7, −7) and (−7, 9)).
3. axis = horizontal, form = vertex, h = 0, k = 0, p = 6 gives focus = (6, 0), directrix = x = −6, ends = (6, −12) and (6, 12), vertexEq = y² = 24x. Source: OpenStax, Algebra and Trigonometry 2e, §12.3 The Parabola (standard forms (x − h)² = 4p(y − k) with focus (h, k + p) and directrix y = k − p, and (y − k)² = 4p(x − h) with focus (h + p, k) and directrix x = h − p; latus rectum endpoints (h ± 2p, k + p) and (h + p, k ± 2p)), https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-3-the-parabola (retrieved 2026-10-02) (Example 1: y² = 24x, p = 6, focus (6, 0), directrix x = −6, endpoints (6, ±12)).
4. axis = vertical, form = standard, a = 2, b = -6, c = 7 gives vertex = (3/2, 5/2), focus = (3/2, 21/8), directrix = y = 19/8, pValue = 0.125, vertexEq = (x − 3/2)² = (1/2)(y − 5/2). Source: OpenStax, Algebra and Trigonometry 2e, §12.3 The Parabola (standard forms (x − h)² = 4p(y − k) with focus (h, k + p) and directrix y = k − p, and (y − k)² = 4p(x − h) with focus (h + p, k) and directrix x = h − p; latus rectum endpoints (h ± 2p, k + p) and (h + p, k ± 2p)), https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-3-the-parabola (retrieved 2026-10-02).

## FAQ

### How do I find the focus of a parabola?

Write it in standard form (x − h)² = 4p(y − k). The vertex is (h, k), and the focus is p units from the vertex along the axis: (h, k + p). For (x − 4)² = 28(y + 8), 4p = 28, so p = 7 and the focus is (4, −8 + 7) = (4, −1).

### How do I find the directrix?

The directrix is the line p units on the other side of the vertex: y = k − p for a parabola that opens up or down, x = h − p for one that opens sideways. For (x − 4)² = 28(y + 8) it is y = −8 − 7 = −15.

### How do I get p from y = ax² + bx + c?

p = 1 ÷ (4a). For y = 2x² − 6x + 7, p = 1/8. The vertex is at x = −b ÷ (2a) = 3/2 and y = c − b² ÷ (4a) = 5/2, so the focus is (3/2, 5/2 + 1/8) = (3/2, 21/8).

### Which way does the parabola open?

It opens toward the focus. For (x − h)² = 4p(y − k) it opens up when p > 0 and down when p < 0. For (y − k)² = 4p(x − h) it opens right when p > 0 and left when p < 0.

### What is the latus rectum?

The chord through the focus parallel to the directrix. Its length is |4p| and its ends are 2p from the focus on each side. For y² = 24x, p = 6, so the endpoints are (6, −12) and (6, 12), 24 apart.

### What makes a curve a parabola?

Every point on it is the same distance from the focus as from the directrix. That is why the vertex sits halfway between them.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §12.3 The Parabola (a parabola is the set of points equidistant from a fixed line, the directrix, and a fixed point, the focus; standard forms (x − h)² = 4p(y − k), focus (h, k + p), directrix y = k − p, latus rectum endpoints (h ± 2p, k + p); (y − k)² = 4p(x − h), focus (h + p, k), directrix x = h − p, endpoints (h + p, k ± 2p); Examples 1, 4 and 5). https://openstax.org/books/algebra-and-trigonometry-2e/pages/12-3-the-parabola (retrieved 2026-10-02)