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Parabola Calculator

Find the vertex, focus, and directrix of a parabola.

Introduction

Parabola Calculator — Find the vertex, focus, and directrix of a parabola. Enter a, b, c to get an instant, accurate result.

Formula

For y = ax² + bx + c: vertex x = −b/(2a), vertex y = a(vertex x)² + b(vertex x) + c. With p = 1/(4a), the focus is at (vertex x, vertex y + p) and the directrix is the line y = vertex y − p.

Step-by-step

  1. Enter the a.
  2. Enter the b.
  3. Enter the c.
  4. Click Calculate to see your result instantly.

Real-world example

Example: With a = 1, b = -4, c = 3, the Parabola Calculator gives Vertex X: 2, Vertex Y: -1, Focus Y: -0.75.

Frequently Asked Questions

What if a = 0?
That's a straight line, not a parabola — a must be nonzero.
What are the focus and directrix used for?
Every point on the parabola is equidistant from the focus and the directrix line — they define the parabola's geometric shape.
What inputs does the Parabola Calculator need?
You'll need to provide: a, b, c. All fields use sensible defaults, so you can see a working example immediately and then adjust the values to match your own numbers.
How accurate is the Parabola Calculator?
The Parabola Calculator applies the formula exactly as calculated — For y = ax² + bx + c: vertex x = −b/(2a), vertex y = a(vertex x)² + b(vertex x) + c. With p = 1/(4a), the focus is at (vertex x, vertex y + p) and the directrix is the line y = vertex y − p. — so results are precise for the inputs you provide. Accuracy depends on entering correct, realistic input values.
Is the Parabola Calculator free to use?
Yes, the Parabola Calculator is completely free, requires no signup, and runs instantly in your browser.

About the Parabola Calculator

The Parabola Calculator uses a real, verifiable formula — For y = ax² + bx + c: vertex x = −b/(2a), vertex y = a(vertex x)² + b(vertex x) + c. With p = 1/(4a), the focus is at (vertex x, vertex y + p) and the directrix is the line y = vertex y − p. — so results are accurate every time, not an approximation.