Parabola Calculator

Free parabola calculator. Convert between y = ax² + bx + c, vertex form, 3-point form, and focus-directrix. Get vertex, focus, directrix, axis of symmetry, roots and discriminant.

Full original guide (expanded)

Parabola Calculator

Solve parabola forms from standard, vertex, three-point, or focus/directrix inputs.

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

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).

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

Parabola Calculator – FAQ

Audit: Complete
Formula (LaTeX) + variables + units
This section shows the formulas used by the calculator engine, plus variable definitions and units.
Formula (extracted LaTeX)
\[','\\]
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Formula (extracted text)
Standard form: \( y = ax^2 + bx + c \) Vertex form: \( y = a(x - h)^2 + k \) Focus-directrix form: \( (x - h)^2 = 4p (y - k) \)
Formula (extracted text)
\( h = -\frac{b}{2a}, \quad k = f(h) = a h^2 + b h + c \)
Formula (extracted text)
If \( y = a(x - h)^2 + k \), then \( p = \frac{1}{4a} \) Focus: \( (h, k + p) \) Directrix: \( y = k - p \)
Variables and units
  • No variables provided in audit spec.
Sources (authoritative):
Changelog
Version: 0.1.0-draft
Last code update: 2026-01-19
0.1.0-draft · 2026-01-19
  • Initial audit spec draft generated from HTML extraction (review required).
  • Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
  • Confirm sources are authoritative and relevant to the calculator methodology.
Verified by Ugo Candido on 2026-01-19
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Formulas

(Formulas preserved from original page content, if present.)

Version 0.1.0-draft
Citations

Add authoritative sources relevant to this calculator (standards bodies, manuals, official docs).

Changelog
  • 0.1.0-draft — 2026-01-19: Initial draft (review required).