The document discusses how to graph hyperbolas when the center is not at the origin. It provides an example of a hyperbola with center (2, -1), which shifts the original graph 2 units right and 1 unit down. It then discusses key features of the hyperbola like vertices, foci, asymptotes, and their locations relative to the shifted center point.

1. Hyperbolas translated away
from the center

2. Example : Suppose the center is not at the origin (0, 0) but is at some
other point such as (2, -1). To graph this hyperbola requires us to
remember how graphs are moved horizontally and vertically by a change
in the equation.
Using Example #1 above, we have
This will move the graph in our previous example 2 units right and 1 unit
down. Both graphs are shown below.

3. Note that a = 3, b = 4,
In this graph the transverse axis is horizontal. Thus each focus is a
distance of 5 horizontally from the center. One focus is at (7, -1) and one is at
(-3, -1). Note that the graphing calculator does not do a good job of showing
the top and bottom halves of the branches of the hyperbola joining at the
vertices which are located at (-3, -1) and (5, -1).
Using this as a model, other equations describing hyperbolas with centers at
(2, -1) can be written.
and the slope of the asymptotes is
If a = 3 and b = 2, and the transverse axis is horizontal, the equation is
The slope of the asymptotes is and the vertices are located
at (-1, -1) and (5, -1).

4. The foci are located at
If a = 2 and b = 4, and the transverse axis is vertical, the equation is