Parabola Focus and Directrix Calculator

How it works

y = ax² + bx + c: vertex (h, k) with h = −b/2a, k = c − b²/4a · p = 1/(4a) · focus (h, k + p) · directrix y = k − p · (x − h)² = 4p(y − k)

A parabola is the set of points equally distant from a fixed point (the focus) and a fixed line (the directrix). Completing the square puts y = ax² + bx + c into vertex form a(x − h)² + k, and comparing with the conic form (x − h)² = 4p(y − k) gives the focal length p = 1/(4a). The focus sits p units from the vertex inside the curve and the directrix p units on the other side; the latus rectum, the chord through the focus parallel to the directrix, has length |4p|. For x = ay² + by + c the roles of x and y swap and the parabola opens sideways.

Worked example

For y = x² − 4x + 3: h = 4/2 = 2 and k = 3 − 16/4 = −1, so the vertex is (2, −1). With a = 1, p = 1/4, so the focus is (2, −1 + 1/4) = (2, −3/4) and the directrix is y = −1 − 1/4 = −5/4. The parabola opens upward, its axis is x = 2 and its 4p form is (x − 2)² = (y + 1).

Sources: Parabola — Wikipedia · Parabola — Math is Fun

Frequently asked questions