This is a rich topic, which I use to introduce sequences. It is also a good opportunity to introduce subscript notation and limits. It is a necessary beginning to any discussion of dynamical systems.
This work can start at almost any level in high school. It can be distributed into Algebra 1, Algebra 2, and Precalculus, or taught as one unit.
Henri Picciotto

Linear
Two packets:
 "Iterating Linear Functions: An Introduction to Dynamical Systems". These are activities which I coauthored with Jonathan Choate for The Mathematics Teacher, February 1997, special issue on Algebra. Use them to introduce these ideas.
 Teacher's Guide (PDF)
 Student Sheets (PDF)
 Some lessons from Algebra: Themes, Tools, Concepts, (1994) the book I coauthored with Anita Wah. You can use "Instant Riches" before the above packet, and the rest as a source of additional problems, homework, and assessments. Two PDF files:
 Teacher's Guide (PDF)
 Student Sheets (PDF)
Note that there is a difference in terminology between the two packets: what is known as a "time series graph" in the first is called "linked function diagrams" in the second.
 Both packets require technological support, such as a spreadsheet program, a graphing calculator, or Fathom:
 Arithmetic (Fathom)
 Linear (Fathom)
As a followup to these lessons, here is a worksheet I use in Infinity, a postAlgebra 2 elective course, to review the above material at a higher level, and to find an explicit formula for the nth term:
Explicit Formula (PDF)
NonLinear
 After this, we are ready for an exploration of iterating some nonlinear functions:
 See Infinity, Unit 2: Chaos
 Finally, a numbertheoretic extension:
 Iterating f(x)=2x2
