Hyperbolas don't come up much — at least
not that I've noticed — in other math classes, but if you're covering
conics, you'll need to know their basics. An hyperbola looks sort of like
two mirrored parabolas, with the two "halves"
being called "branches". Like an ellipse,
an hyperbola has two foci and two vertices; unlike an ellipse, the foci
in an hyperbola are further from the hyperbola's center than are its vertices:

The hyperbola
is centered on a point (h,
k), which is the "center"
of the hyperbola. The point on each branch closest to the center is that
branch's "vertex".
The vertices are some fixed distance a
from the center. The line going from one vertex, through the center, and
ending at the other vertex is called the "transverse" axis.
The "foci"
of an hyperbola are "inside" each branch, and each focus is
located some fixed distance c
from the center. (This means that a
< c for hyperbolas.)
The values of a and
c will
vary from one hyperbola to another, but they will be fixed values for
any given hyperbola.

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For any point on an ellipse, the sum
of the distances from that point to each of the foci is some fixed value;
for any point on an hyperbola, it's the difference of the distances
from the two foci that is fixed. Looking at the graph above and letting
"the point" be one of the vertices, this fixed distance must
be (the distance to the further focus) less (the distance to the nearer
focus), or (a + c)
– (c – a) = 2a.
This fixed-difference property can used for determining locations: If
two beacons are placed in known and fixed positions, the difference in
the times at which their signals are received by, say, a ship at sea can
tell the crew where they are.

As with ellipses, there is a relationship
between a,
b,
and c,
and, as with ellipses, the computations are long and painful. So trust
me that, for hyperbolas (where a
< c), the relationship is
c^{2} – a^{2}
= b^{2} or, which means
the same thing, c^{2}
= b^{2} + a^{2}.
(Yes, the Pythagorean Theorem is used to prove this relationship. Yes,
these are the same letters as are used in the Pythagorean Theorem. No,
this is not the same thing as the Pythagorean Theorem. Yes, this is very
confusing. Just memorize it, and move on.)

When the transverse axis is horizontal
(in other words, when the center, foci, and vertices line up side
by side, parallel to the x-axis),
then the a^{2} goes
with the x part
of the hyperbola's equation, and the y part
is subtracted.

When the transverse axis is vertical
(in other words, when the center, foci, and vertices line up above
and below each other, parallel to the y-axis),
then the a^{2} goes
with the y part
of the hyperbola's equation, and the x part
is subtracted.

In "conics" form, an hyperbola's
equation is always "=1".

For reasons you'll learn in calculus, the
graph of an hyperbola gets fairly flat and straight when it gets far away
from its center. If you "zoom out" from the graph, it will look
very much like an "X", with maybe a little curviness near the
middle. These "nearly straight" parts get very close to what
are called the "asymptotes" of the
hyperbola. For an hyperbola centered at (h,
k) and having fixed values a
and b,
the asymptotes are given by the following equations:

hyperbolas'
graphs

asymptotes'
equations

Note that the only difference in the asymptote
equations above is in the slopes of the straight lines: If a^{2}
is the denominator for the x
part of the hyperbola's equation, then a
is still in the denominator in the slope of the asymptotes' equations;
if a^{2}
goes with the y
part of the hyperbola's equation, then a
goes in the numerator of the slope in the asymptotes' equations.

Hyperbolas can be fairly "straight"
or else pretty "bendy":

hyperbola
with an eccentricity of about 1.05

hyperbola
with an eccentricity of about 7.6

The measure of the amount of curvature
is the "eccentricity" e,
where e = c/a.
Since the foci are further from the center of an hyperbola than are the
vertices (so c >
a for hyperbolas), then e
> 1. Bigger values of e
correspond to the "straighter" types of hyperbolas, while values
closer to 1
correspond to hyperbolas whose graphs curve quickly away from their centers.