Parabola Calculator — Vertex, Focus & Directrix

Enter the coefficients of y = ax² + bx + c to find the parabola’s vertex, focus, directrix, axis of symmetry and vertex form. The free calculator also draws the graph and shows the calculation steps. It handles parabolas that open up or down; a must not be zero.

Parabola calculator

Enter your equation in coefficient form

First arrange the equation as y = ax² + bx + c. Enter the signed number multiplying x² in a, the signed number multiplying x in b, and the constant in c. Use 0 for a missing x term or constant. For y = −2x² + 8x − 3, enter −2, 8 and −3, then press Analyze.

The fields accept numbers and decimals. For a coefficient such as 1/2, enter 0.5. If your equation is y = a(x − h)² + k, expand it first or use the completing the square calculator to check the conversion between coefficient and vertex forms.

An equation with a = 0 is a line, so the calculator rejects it. A sideways parabola such as x = y² is outside this tool’s input format. Use the full-screen link if you want a larger working area. Displayed values are rounded to four decimal places; keep fractions in your written solution when an exact answer is required.

Read the vertex, focus and directrix

For y = ax² + bx + c, calculate h = −b/(2a), then k = ah² + bh + c. The vertex is (h, k), and the vertical axis of symmetry is x = h. Vertex form is y = a(x − h)² + k.

Set p = 1/(4a). The focus is (h, k + p), and the directrix is y = k − p. Here p is a signed vertical offset: it is positive for an upward-opening parabola and negative for a downward-opening one. The actual distance from the vertex to the focus or directrix is |p|, which is always positive.

The focus lies inside the opening; the directrix lies on the opposite side of the vertex. Every point on the parabola is equally far from the focus and the directrix. These relationships follow the standard forms explained in OpenStax’s parabola lesson.

Two worked examples to check your setup

Upward opening: y = x² − 4x + 3

Enter a = 1, b = −4 and c = 3. Then h = −(−4)/(2 × 1) = 2, and k = 2² − 4(2) + 3 = −1. Completing the square gives y = (x − 2)² − 1.

The vertex is (2, −1), the axis is x = 2 and p = 0.25. The focus is (2, −0.75), while the directrix is y = −1.25. Both are 0.25 unit from the vertex. Check the graph by substituting x = 1 and x = 3: both give y = 0, on opposite sides of the axis.

Downward opening: y = −2x² + 8x − 3

Enter a = −2, b = 8 and c = −3. Here h = 2, k = 5 and y = −2(x − 2)² + 5. The vertex is (2, 5), and p = −0.125.

The focus is (2, 4.875) and the directrix is y = 5.125. The focus is below the vertex because the parabola opens downward. The distance is 0.125 unit, not −0.125. Substituting x = 2 into the original equation gives 5, confirming the vertex’s height.

Practice the graph and review a mistake

Before calculating, predict the opening from the sign of a. Afterward, check that the focus and directrix fall on opposite sides of the vertex and that two x-values equally spaced from h give the same y-value. The vertex is a minimum when a > 0 and a maximum when a < 0; it is not necessarily an x-intercept.

If the graph or signs are unfamiliar, work through the quadratic graphing lesson and worksheet. Use the Algebra 2 learning center for related topic review, or browse the existing free math calculators when you need a different tool. Solve a similar problem on paper afterward to check that you can reproduce the method.