This is often called the logistic function. Discrete Dynamical Systems may have orbits that cycle with any period. In general, the orbit of lies on a cycle of period n if and n is the smallest positive integer for which this happens. We will study this type of behavior later. Finding Fixed Points.
Lets confirm our work above geometrically.
This is geometrically confirmed below. We saw previously that the fixed points for F are 2 and -1. As you might guess, the algebra here is not trivial. Two of those are fixed points for F, the other six are periodic with period 3. In other words there are two orbits, each of period three.