Parabola Calculator

Analyse a parabola y = ax² + bx + c — vertex, axis of symmetry, focus, directrix and intercepts — with everything marked on a graph.

y = x2 + x +
Vertex—
Axis of symmetry—
Focus—
Directrix—
x-intercepts—
y-intercept—

How to Analyse a Parabola

Enter the coefficients a, b and c to get the vertex, focus, directrix and intercepts.

1

Enter the quadratic

Type the coefficients of y = ax² + bx + c. a must not be zero for the graph to be a parabola.

2

Read the key features

The vertex is at x = −b ⁄ (2a); the axis of symmetry passes through it. The focus sits 1 ⁄ (4a) above the vertex and the directrix the same distance below.

3

Read the intercepts

The x-intercepts are the real roots of the quadratic (there may be two, one or none), and the y-intercept is (0, c).

Frequently Asked Questions

How do you find the focus and directrix of a parabola?
For y = ax² + bx + c with vertex (h, k), the focus is (h, k + 1 ⁄ (4a)) and the directrix is the line y = k − 1 ⁄ (4a). The focus is inside the curve and the directrix is the mirror line the same distance away. Every point on the parabola is equally distant from the focus and the directrix, which is the geometric definition of a parabola.
What are the x-intercepts of a parabola?
The x-intercepts are the x-values where y = 0 — the real roots of ax² + bx + c. A parabola can cross the x-axis twice, touch it once, or not at all, depending on the discriminant b² − 4ac. The calculator finds them by solving ax² + bx + c = 0 with the quadratic formula, so they may come back as a pair, a single repeated value, or a note that no real intercepts exist.
What is the y-intercept of a parabola?
The y-intercept is where x = 0, which gives y = c. So the parabola always crosses the y-axis at (0, c). This is the easiest intercept to find, since it only requires plugging in x = 0 rather than solving an equation.
What is the axis of symmetry?
The axis of symmetry is the vertical line x = −b ⁄ (2a) through the vertex; the parabola is symmetric across it. Every point on one side of the axis has a mirror-image point on the other side at the same height. Because the vertex sits on this line, finding the axis is often the fastest way to locate the vertex's x-coordinate.
Is my data sent anywhere?
No — the parabola calculator runs entirely in your browser using JavaScript, with no server round-trip. Your coefficients are never uploaded, logged or stored. The tool keeps working without an internet connection.

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