Parabola and its standard forms

A parabola is the set of all points of a plane that are equally far from a fixed line, the directrix, and a fixed point not on it, the focus. The name is often read as 'para' (for) and 'bola' (throwing): the path of a thrown ball.

If the fixed point lay on the fixed line, the equidistant points would form the straight line through that point perpendicular to the fixed line: a degenerate parabola.

The axis is the line through the focus perpendicular to the directrix. The vertex is where the parabola meets its axis, midway between the focus and the directrix.

Put the vertex at the origin and the focus at (a, 0) with a > 0, so the directrix is x = −a. For P(x, y), PF² = (x − a)² + y² and the distance to the directrix is |x + a|. Setting them equal and squaring gives y² = 4ax, and every point of y² = 4ax satisfies PF = x + a in return.

The four standard forms: y² = 4ax (opens right), y² = −4ax (opens left), x² = 4ay (opens up), x² = −4ay (opens down), with a > 0 in each. In every one the vertex is the origin and the focus lies on an axis at distance a.

Reading a standard equation: a y² term means the axis of symmetry is the x-axis; an x² term means it is the y-axis. The sign of the first-degree term gives the direction of opening.

In y² = 4ax, x can never be negative, so the curve lies in the first and fourth quadrants and runs on without end, symmetric about the x-axis.

Parabolas with a vertex elsewhere, or an axis not along a coordinate axis, are outside this chapter.