How do I find the vertex of a parabola?

If the parabola is y = ax² + bx + c, the vertex is at x = -b/(2a) and y = f(x). The calculator does this automatically.

Can I define a parabola from 3 points?

Yes. Enter three distinct points (x, y) and the tool will solve for a, b, c so you get y = ax² + bx + c passing through all three.

What if I have focus and directrix?

Enter focus (x_f, y_f) and a horizontal directrix y = d. The tool will find the vertex, parameter p, and rewrite the parabola in standard form.

Parabola Calculator

Define a parabola from the form you already have – standard, vertex, 3 points, or focus & directrix – and get every other representation instantly: vertex, focus, directrix, axis, roots, discriminant.

Multi-form Shows steps (key values) Classroom-friendly

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Decimals

Orientation

This version solves vertical parabolas.

Parabola basics

A vertical parabola can be written in several equivalent ways:

Standard form: \( y = ax^2 + bx + c \)

Vertex form: \( y = a(x - h)^2 + k \)

Focus-directrix form: \( (x - h)^2 = 4p (y - k) \)

From standard form you can always find the vertex:

\( h = -\frac{b}{2a}, \quad k = f(h) = a h^2 + b h + c \)

And from vertex form you can always find the focus and directrix:

If \( y = a(x - h)^2 + k \), then \( p = \frac{1}{4a} \)

Focus: \( (h, k + p) \)

Directrix: \( y = k - p \)

Example: parabola through 3 points

Suppose the parabola passes through (0, 2), (1, 0) and (2, 6).

Plug each point into \( y = ax^2 + bx + c \) to get 3 equations.
Solve the system to get a, b, c.
The calculator does this instantly and then gives you vertex, focus and roots.

Parabola basics

Example: parabola through 3 points

Parabola Calculator – FAQ