Definition

A parabola is the set of all points that are the same distance from a fixed line (called the directrix) and a fixed point (focus) not on the directrix. See Figure P1 for the view of a parabola and its related focus and directrix.

Note that the graph of a parabola is similar to one branch of a hyperbola. However, you should realize that a parabola is not simply one branch of a hyperbola. Indeed, the branches of a hyperbola approach linear asymptotes, while a parabola does not do so.

Several other terms exist which are associated with a parabola. The midpoint between the focus and directrix of the parabola is called the vertex and the line passing through the focus and vertex is called the axis of the parabola. (This is similar to the major axis of the ellipse and the transverse axis of the hyperbola.) See Figure P2.

Now let's move to the standard algebraic equations for parabolas and note the four types of parabolas that exist. As we discuss the four types, you should notice the differences in the equations that are related to each of the four parabolas.

The standard form of the equation of the parabola with vertex at (0,0) with the focus lying d units from the vertex is given by

if the axis is vertical and

if the axis is horizontal. See Figure P3 for an example with vertical axis and Figure P4 for an example with horizontal axis.

Note here that we have assumed that

It is also the case that d could be negative, which flips the orientation of the parabola. See Figures P5 and P6.

Thus, we see that there are four different orientations of parabolas, which depend on a) which variable is squared (x or y) and b) whether d is positive or negative.

One last comment before going to some examples. If the vertex of the parabola is at , then the equation of the parabola does change slightly. The equation of a parabola with vertex at is given by

if the axis is vertical and

if the axis is horizontal. Now let's move to some specific examples.