Parabola calculator

How do I find the vertex, focus and directrix of a parabola?

Write the equation in vertex form y = a(x − h)² + k. The vertex is (h, k), the distance from the vertex to the focus is p = 1 ÷ 4a, the focus is (h, k + p) and the directrix is the line y = k − p. For y = x² − 4x + 3 = (x − 2)² − 1: vertex (2, −1), p = 0.25, focus (2, −0.75), directrix y = −1.25.

Example: x² − 8x − 28y − 208 = 0

Completing the square gives (x − 4)² = 28(y + 8), so 4p = 28 and p = 7. The vertex is (4, −8), the axis of symmetry x = 4, the focus (4, −1), the directrix y = −15, and the latus rectum runs from (−10, −1) to (18, −1). This is Example 5 in OpenStax Precalculus 2e, section 10.3.

What is the latus rectum (focal width)?

The chord through the focus, parallel to the directrix. Its length is |4p| = 1 ÷ |a|, and its ends are 2p on either side of the focus. It shows how wide the parabola is at the height of the focus.

Which way does a parabola open?

y = ax² + bx + c opens up when a > 0 and down when a < 0. When y is the squared variable — x = ay² + by + c — it opens right for a > 0 and left for a < 0. For (y − 1)² = −16(x + 3): 4p = −16, p = −4, so it opens left, with vertex (−3, 1), focus (−7, 1) and directrix x = 1 (OpenStax Example 4).

How is this different from the quadratic formula calculator?

The quadratic formula calculator solves ax² + bx + c = 0 for its roots, complex ones included. This calculator treats the equation as a curve: the vertex, focus, directrix and focal width, the intercepts with both axes, and the same parabola in standard, vertex and focus form.

Where do the formulas come from?

From OpenStax Precalculus 2e, section 10.3 “The Parabola”: the standard forms (x − h)² = 4p(y − k) and (y − k)² = 4p(x − h), with focus, directrix and latus rectum. Its worked examples are part of this calculator’s automated tests.

How do I find the vertex from y = ax² + bx + c?

The x-coordinate of the vertex is −b ÷ 2a, and the y-coordinate comes from substituting it back (or from c − b² ÷ 4a). For y = 2x² − 12x + 22 that gives x = 3 and y = 4, so the vertex is (3, 4).

A worked example from the general form

Take y = 2x² − 12x + 22. Here a = 2, b = −12 and c = 22. The vertex is at x = −(−12) ÷ (2 × 2) = 3, and y = 22 − 144 ÷ 8 = 4. In vertex form the equation is y = 2(x − 3)² + 4. The focal distance is p = 1 ÷ (4 × 2) = 0.125, so the focus is (3, 4.125) and the directrix is the line y = 3.875. The latus rectum has length 4p = 0.5. Because a is positive the parabola opens up, and the y-intercept is c = 22.

A parabola that opens sideways

If y is the squared variable, the roles of x and y swap. For x = −0.5y² + 2y + 1 the vertex is (3, 2), the axis is the horizontal line y = 2, p = −0.5, and since a is negative the curve opens left. The focus is (2.5, 2) and the directrix is the vertical line x = 3.5. Every point on the curve is the same distance from the focus as from the directrix, which is the definition of a parabola.

Common mistakes

Mixing up p and a is the most frequent slip: in vertex form y = a(x − h)² + k, p = 1 ÷ 4a, whereas in the focus form (x − h)² = 4p(y − k) the number in front of the bracket is 4p itself. Sign errors in the vertex are the next: (x + 3)² means h = −3. Finally, the focus lies inside the curve on the side it opens toward and the directrix on the opposite side of the vertex, both a distance |p| away; if your focus ends up outside the curve, the sign of p is wrong.

Sources: OpenStax Precalculus 2e, section 10.3 (The Parabola) for the standard forms, focus, directrix and latus rectum; the vertex formula from OpenStax College Algebra 2e on quadratic functions. Related: quadratic formula calculator, completing the square calculator, ellipse calculator and hyperbola calculator.