Parabolas

Conic sections are curves formed when a plane intersects a double-napped cone. If the plane runs parallel to the cone’s slant, the resulting curve is a parabola.

A parabola is a set of points equidistant from a fixed point—the focus—and a fixed line—the directrix.

The axis of symmetry passes through the vertex. The focus lies along this axis, while the directrix is perpendicular to it on the opposite side.

The standard form of a parabola arises from its geometric definition, where the distances to the focus and the directrix are equal.

Applying the distance formula to both distances and squaring each side eliminates the square root.

Expanding both expressions and simplifying removes common terms and reveals the standard form when the axis is vertical.

Swapping x and y yields the standard form for a horizontal axis. Shifting the vertex from the origin further modifies the equation.

The parabola opens upward or to the right if the focus lies in the positive direction from the vertex. It opens downward or to the left if the focus lies in the negative direction.

These parabolic forms and orientations appear in structures, like suspension bridges or satellite dishes.