As mentioned above, a function in the class of quadratic rational functions with a repelling fixed point at zero can have zero, one or two attracting fixed points or cycles. We now break these into six groups:
no attracting fixed points or cycles
one attracting fixed point and one attracting fixed cycle
We will see that each group contains a variety of interesting graphs and that the ways in which the points and cycles interact with each other are diverse. I suspect that these patterns of interaction in the graph of G are saying something profound about f but I have been able to discover very little about what is being said.