The "basic" hyperbola is the function
This function has a graph which consists of two disjoint parts. Note that 0 is not in the domain. Note also that the function approaches 0 asymptotically as x grows infinitely large (or infinitely negative), and that it approaches infinity as x approaches 0 from the positive side, negative infinity as x approaches 0 from the negative side.
The center of the hyperbola is the point of rotational symmetry. In the above example, the center is (0, 0).
The graph of the rational function
is a hyperbola whose center is (h, k). The constant a controls the "steepness" with which the graph approaches the asymptotes.