Vertical Parabola
Vertex form: y = a(x − h)² + k
Focus: (h, k + 1/(4a))
Directrix: y = k − 1/(4a)
Horizontal Parabola
Vertex form: x = a(y − k)² + h
Focus: (h + 1/(4a), k)
Directrix: x = h − 1/(4a)
Vertex + Focus: Enter vertex coordinates and focus point. Focus must be aligned with vertex.
Vertex + Directrix: Enter vertex and the directrix line value.
Standard Form: Enter coefficients a, b, c for the equation.
A parabola is a U-shaped curve that is the graph of a quadratic function. It is defined as the set of all points that are equidistant from a fixed point called the focus and a fixed line called the directrix. Parabolas have many important applications in physics, engineering, and architecture, from satellite dishes to the paths of projectiles.
The vertex of a parabola is the point where it changes direction, and it represents either the minimum or maximum value of the quadratic function depending on whether the parabola opens upward or downward. The axis of symmetry passes through the vertex and divides the parabola into two mirror-image halves.
Parabola calculations follow standard mathematical formulas. Results depend on correct input values and orientation. This calculator provides approximate decimal representations where exact values may involve irrational numbers.