# Parabola Calculator

Calculate the vertex, focus, directrix, and axis of symmetry for any parabola. Supports standard form and vertex form. Free parabola calculator.

## What this calculates

Find the vertex, focus, directrix, axis of symmetry, and focal length for any parabola. Enter the equation in standard form (y = ax² + bx + c) or vertex form (y = a(x - h)² + k) and get all key properties instantly.

## Inputs

- **Equation Form** — options: Standard: y = ax² + bx + c, Vertex: y = a(x - h)² + k — Choose the form of your parabola equation.
- **Coefficient a** — Controls the width and direction of the parabola. Must not be zero.
- **Coefficient b** — The coefficient of x in standard form.
- **Coefficient c** — The constant term in standard form.
- **Vertex h (x-coordinate)** — The x-coordinate of the vertex in vertex form.
- **Vertex k (y-coordinate)** — The y-coordinate of the vertex in vertex form.

## Outputs

- **Vertex (h, k)** — formatted as text — The turning point of the parabola.
- **Focus** — formatted as text — The focus point of the parabola.
- **Directrix** — formatted as text — The directrix line equation.
- **Axis of Symmetry** — formatted as text — The vertical line through the vertex.
- **Opens** — formatted as text — Whether the parabola opens upward or downward.
- **Focal Length** — The distance from the vertex to the focus: 1/(4|a|).

## Details

A parabola is a U-shaped curve defined by a quadratic equation. Every parabola has a vertex (turning point), a focus (a special interior point), and a directrix (a line outside the curve). Any point on the parabola is the same distance from the focus as it is from the directrix.

**Key Properties:**

- **Vertex:** The highest or lowest point, located at (h, k). In standard form, h = -b/(2a) and k = c - b²/(4a).
- **Focus:** Located at (h, k + 1/(4a)). This is the point where reflected rays converge, which is why satellite dishes and headlights use parabolic shapes.
- **Directrix:** A horizontal line at y = k - 1/(4a), always on the opposite side of the vertex from the focus.
- **Axis of symmetry:** The vertical line x = h that passes through both the vertex and focus.
- **Focal length:** The distance from the vertex to the focus, equal to 1/(4|a|). A smaller |a| means a wider parabola with a longer focal length.

**Standard vs. Vertex Form:**

Standard form y = ax² + bx + c is useful for identifying intercepts. Vertex form y = a(x - h)² + k directly reveals the vertex and makes graphing easier. Converting between them involves completing the square.

## Frequently Asked Questions

**Q: How do I find the vertex of a parabola?**

A: For the standard form y = ax² + bx + c, the vertex is at x = -b/(2a). Plug that x back in to get y. In vertex form y = a(x - h)² + k, the vertex is simply the point (h, k). The vertex represents the minimum value when a > 0 and the maximum when a < 0.

**Q: What is the focus of a parabola used for?**

A: The focus is the point where all parallel rays reflecting off the parabola converge. This property is used in satellite dishes (signals focus at one point), headlights (light from the focus reflects outward in parallel), and solar concentrators. The focus sits at distance 1/(4|a|) from the vertex.

**Q: What is the directrix?**

A: The directrix is a line perpendicular to the axis of symmetry, located on the opposite side of the vertex from the focus. The defining property of a parabola is that every point on the curve is equidistant from the focus and the directrix. For y = ax² + bx + c, the directrix is the line y = k - 1/(4a).

**Q: Does a larger or smaller value of a make the parabola wider?**

A: A smaller absolute value of a makes the parabola wider, while a larger absolute value makes it narrower. When |a| = 1, you get a standard-width parabola. When |a|  1, the curve is more compressed. The sign of a determines whether the parabola opens up (positive) or down (negative).

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Source: https://vastcalc.com/calculators/math/parabola
Category: Math
Last updated: 2026-04-08