Given an initial value x_{0} and a function f, we generate the sequence 
x_{0}, x_{1} = f(x_{0}),
x_{2} = f(x_{1}), x_{3} = f(x_{2}), ... 
The algebraic mechanism is clear, though perhaps unenlightening. 
Graphical iteration is a geometrical
method for visualizing this process. 
Briefly, we use an alternating collection of
vertical and horizontal segments to connect these points. 

(x_{0},0) to (x_{0},f(x_{0})) = (x_{0},x_{1}) 
(x_{0},x_{1}) to (x_{1},x_{1}) 
(x_{1},x_{1}) to (x_{1},f(x_{1})) =
(x_{1},x_{2}) 
(x_{1},x_{2}) to (x_{2},x_{2}) 
and so on. 

Here is an illustration of the first few steps
of graphical iteration. 
Here is an example of sensitivity to initial conditions for the
tent map. 