Parabola Calculator
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.
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.