Geometric Definition and Equations of Parabolas

A parabola is the set of points in a plane equidistant from a fixed point, called focus, and fixed line, called directrix, and where the axis of symmetry passes through the focus and is perpendicular to the directrix.  

Moreover, the vertex is the midpoint of the line segment joining the focus and the directrix.

An equation of a parabola may be obtained from the definition above. Here's the development:

Look at the four types of parabolas:

Parabola with Vertical Axis of Symmetry and Vertex (0,0) with focus (0, p) and directrix y = – p

has equation x² = 4py.  

The parabola has vertical axis of symmetry x = 0 and opens upward if p > 0 or opens downward if p < 0.

Parabola with Horizontal Axis of Symmetry and Vertex (0,0) with focus (p, 0) and directrix x = – p

has equation y² = 4px.   The parabola has vertical axis of symmetry y = 0 and opens to the right if p > 0 or opens to the left if p < 0.

Equation Forms for Translated Parabola with vertex (h, k) has an equation of one of the following forms:

Vertical Parabola

Horizontal Parabola

The focus is distance | p | from the vertex.