Note that this is a valid equation for a hyperbola, even though it is not in standard form. Our first goal is to rewrite this equation into standard form and then to interpret this equation as we sketch the graph. As we have seen in previous work, we need to use the technique of ``completing the square'' to work this out.
We see that
is equivalent to
Now we want to fill in the apparent gaps that have been inserted in the parentheses above. This ``filling in'' is completing the square. We want to write in the number that will make each set of parentheses a perfect square. We do that now:
Note that, when we add 9 to the left--hand side of the equation, we must also add it to the right--hand side. Also, we are not really adding 1 to the left--hand side; we are really subtracting 4 since we multiply the 1 by the -4 that is outside the parentheses. Hence, we must also subtract 4 from the right--hand side.
Rewriting our equation now yields
or
and we have successfully transformed the equation originally given to us into
the standard equation of a hyperbola. This hyperbola has center 
and
has values
and
Since the variable y is in the ``positive'' term here, we know that the transverse axis of this hyperbola is vertical (parallel with the y--axis). Thus, the equations of the asymptotes are
and
The graph of this hyperbola is given in Figure H8.
James A. Sellers