The general

equation for a

hyperbola is

__(x-h)__^{2} - __(y-k)__^{2} = +/-1
a^{2} b^{2}

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
a^{2} b^{2}

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 = a^{2} + b^{2}

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:

ax^{2} + 2hxy + by^{2} + fx + gy + c = 0.

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

ab - h^{2} > 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.