acalculator

What is the focus of my parabola?

Type a parabola as y = ax² + bx + c (or x = ay² + by + c when it opens sideways), or by its vertex and p. The parabola calculator gives the vertex, focus, directrix, axis of symmetry and latus rectum as exact fractions, both forms of the equation, and a graph.

Your numbers

Opens
I know
Focus
(3/2, 21/8)

The parabola y = 2x² − 6x + 7 has its focus at (3/2, 21/8) and its directrix at y = 19/8.

Vertex
(3/2, 5/2)
Directrix
y = 19/8
Axis of symmetry
x = 3/2
Opens
Up
p
0.125
Latus rectum length
0.5
Latus rectum endpoints
(5/4, 21/8) and (7/4, 21/8)
Equation in a, b, c
y = 2x² − 6x + 7
Standard form
(x − 3/2)² = (1/2)(y − 5/2)
h (vertex x)
1.5
k (vertex y)
2.5

Focus: (3/2, 21/8). The parabola y = 2x² − 6x + 7 has its focus at (3/2, 21/8) and its directrix at y = 19/8.

The parabola along its axis, with the vertex marked

How to calculate

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.

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.

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).

  • 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.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Opens Up or down, I know Vertex and p, h 4, k -8, p 7 gives Vertex (4, −8), Focus (4, −1), Directrix y = −15, Axis of symmetry x = 4, Latus rectum endpoints (−10, −1) and (18, −1), Standard form (x − 4)² = 28(y + 8), Equation in a, b, c 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. Opens Left or right, I know Vertex and p, h -3, k 1, p -4 gives Focus (−7, 1), Directrix x = 1, Axis of symmetry y = 1, Opens Left, Latus rectum endpoints (−7, −7) and (−7, 9), Standard form (y − 1)² = −16(x + 3), Latus rectum length 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. Opens Left or right, I know Vertex and p, h 0, k 0, p 6 gives Focus (6, 0), Directrix x = −6, Latus rectum endpoints (6, −12) and (6, 12), Standard form 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. Opens Up or down, I know a, b, c, a 2, b -6, c 7 gives Vertex (3/2, 5/2), Focus (3/2, 21/8), Directrix y = 19/8, p 0.125, Standard form (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)

How it works

Up or down (the axis is vertical): standard form (x − h)² = 4p(y − k).

  • vertex (h, k); focus (h, k + p); directrix y = k − p; axis x = h
  • opens up when p > 0, down when p < 0
  • latus rectum length |4p|, endpoints (h − 2|p|, k + p) and (h + 2|p|, k + p)

Left or right (the axis is horizontal): standard form (y − k)² = 4p(x − h).

  • vertex (h, k); focus (h + p, k); directrix x = h − p; axis y = k
  • opens right when p > 0, left when p < 0
  • latus rectum length |4p|, endpoints (h + p, k − 2|p|) and (h + p, k + 2|p|)

From a, b, c. For y = ax² + bx + c: h = −b ÷ (2a), k = c − b² ÷ (4a), p = 1 ÷ (4a). For x = ay² + by + c the roles swap: k = −b ÷ (2a), h = c − b² ÷ (4a), p = 1 ÷ (4a).

From the vertex and p. a = 1 ÷ (4p). Up or down: b = −2ah and c = ah² + k. Left or right: b = −2ak and c = ak² + h.

Rules

  • a and p may not be 0 (with a = 0 the equation is a line; with p = 0 the focus is on the vertex). Each number is from −1,000,000 to 1,000,000, read as the exact decimal typed, so every answer is an exact fraction.
  • When h, k or 4p is beyond about 1.8 × 10³⁰⁸ (a very close to 0), there is no answer.

Output format. Points show as (x, y) and lines as y = … or x = …, each number a fraction in lowest terms (21/8) or a whole number, with a true minus sign; a fraction longer than 30 characters shows as a decimal with 10 significant digits. "Opens" reads Up, Down, Left or Right. The equation in a, b, c leaves out zero terms, does not write a coefficient of 1, and puts a fraction before a variable in brackets: y = (1/28)x² − (2/7)x − 52/7. The standard form writes 4p before the bracket the same way (a 4p of 1 is not written, −1 is written −): (x − 4)² = 28(y + 8), (x − 3/2)² = (1/2)(y − 5/2), (y − 1)² = (x − 2), x² = −y, y² = 24x. p, the latus rectum length and the vertex decimals show up to 10 significant digits.

Graph. The curve is drawn along the axis from 8|p| before the vertex to 8|p| after it, with the vertex marked. For a sideways parabola the horizontal axis of the graph is y and the vertical axis is x.

Worked examples by hand

(x − 4)² = 28(y + 8) (OpenStax Example 5, from x² − 8x − 28y − 208 = 0). h = 4, k = −8, 4p = 28 so p = 7. Focus (4, −1), directrix y = −15, axis x = 4, latus rectum endpoints (4 ∓ 14, −1) = (−10, −1) and (18, −1). In a, b, c: a = 1/28, b = −2 × (1/28) × 4 = −2/7, c = 16/28 − 8 = −52/7.

(y − 1)² = −16(x + 3) (Example 4). h = −3, k = 1, p = −4: opens left. Focus (−3 − 4, 1) = (−7, 1), directrix x = −3 + 4 = 1, endpoints (−7, 1 ∓ 8) = (−7, −7) and (−7, 9), latus rectum 16.

y² = 24x (Example 1). h = k = 0, p = 6. Focus (6, 0), directrix x = −6, endpoints (6, −12) and (6, 12).

y = 2x² − 6x + 7. h = 6 ÷ 4 = 3/2, k = 7 − 36 ÷ 8 = 5/2, p = 1 ÷ 8. Focus (3/2, 5/2 + 1/8) = (3/2, 21/8), directrix y = 5/2 − 1/8 = 19/8. Standard form (x − 3/2)² = (1/2)(y − 5/2).

Other questions people ask

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.