hyperbola (thing)

The general equation for a hyperbola is

(x-h)2 - (y-k)2 = +/-1
  a2       b2

In this form (h,k) will represent the center of the hyperbola.
If the 1 on the right side of the equation is positive, the equation will represent a hyperbola whose transverse axis is horizontal, looking something like:

\      /
 \    /
  |  |
 /    \
/      \

In this case, the vertices of the hyperbola (i.e the points on the transverse axis, represented in the diagram above by the | symbol) will be (h + a, k) and (h - a, k).

If, on the other hand, the 1 is negative, the hyperbola will have a vertical transverse axis, looking like:

\   /
 \_/
  _
 / \
/   \

In that case, the vertices of the hyperbola will be (h, k + b) and (h, k - b).
The asymptotes of a hyperbola are the lines that the hyperbola approaches; for the hyperbola depicted above, they will look like this:

\   /
 \ /
  X
 / \
/   \

These asymptotes can be found by setting the right side of the equation above to 0, and isolating y:

(x-h)2 - (y-k)2 = 0
  a2       b2

giving:

y = (b/a)x + (k - bh/a)
y = -(b/a)x + (k + bh/a)

By drawing the vertices and the asymptotes of a hyperbola, an approximate sketch of it may be done. Other points of interest include the focii of the hyperbola. The hyperbola itself is the locus of all points such that the difference in their distances from the two focii is constant.
The focii for a hyperbola with the centre at the origin are (c,0) and (-c,0), or (0,c) and (0,-c) for a vertical hyperbola, where

c = a² + b²

Other forms of the equation for a hyperbola exist, as noted in the writeup above. These are all actually transformations of a single, universal form, given by:

ax² + 2hxy + by² + fx + gy + c = 0.

This is a standard form of the equation for all conic sections; when the curve is a hyperbola,

ab - h² > 0

This equation represents hyperbolae rotated to any angle, while the form described above and the form xy=c only denote hyperbolas that are vertical, horizontal, or oriented at 45 degrees to the axes.

Other than annoying students through the vast amounts of work required to graph and rotate these muthas, hyperbolas seem to have no useful application.