NOTE: Enlarge your screen to allow pictures to be two per line! Also, please be seated...these pictures are outstanding!!

Consider the graphs of the equations

When a = 1, b = 1, c = 1, the graph is a hyperbola with asymptotes at x = 1, y = 1.

Look at a few graphs with c = 2, 4, 6.

Now, going in the opposite direction, some interesting things start happening.

Consider what is happening to the equation algebraically.

Solve this for y to get a better feel for the function.

Dividing these to isolate the x:

Now, looking at this equation, things become a bit clearer.

When x = b, we have an asymptote because that fraction would be undefined (and therefore cannot occur for this function).

If c + ab = 0, the equation is y = a which is just a vertical line, no longer a hyperbola. This is the other asymptote.

When c + ab is positive, you have a family of curves in quadrant I and III.

When c + ab is negative, you have a family of curves in quadrant II and IV.

Here is a recap with pictures.