parabola

A parabola is the smooth U-shaped curve you get when you graph a quadratic function. You see it everywhere in the real world: the path of a thrown ball, the cross-section of a satellite dish, the stream from a drinking fountain. It is perfectly symmetric — fold it along its center line and the two halves match exactly.

The graph of f(x) = ax^2 + bx + c is always a parabola. It opens upward if a > 0 and downward if a < 0, and a larger |a| makes it narrower while a smaller |a| makes it wider. Its turning point is the vertex, and the vertical line through the vertex is the axis of symmetry.

Geometrically, a parabola can also be defined without algebra: it is the set of all points equally distant from a fixed point (the focus) and a fixed line (the directrix). This focusing property is why parabolic mirrors and dishes gather parallel rays to a single point — the basis of telescopes and headlights.