Iteration Linear Functions

When we introduce linear equations in our algebra courses we often introduce them in different forms.

Three of the most common are:

-Slope intercept Form: y = mx + b
-Standard Form: ax + by + c = 0
-Point Slope Form: y - y1 = m(x - x1)

I would like to propose a fourth form, Fixed Point Form Iterating linear functions has become a very useful technique. When you iterate a linear function the function's fixed point is often useful.

Definition. F is said to be a fixed point for a function f(x) if f[F] = F.The following Theorem shows how to calculate the fixed point of a linear transformation.

Theorem: The function f[x] = ax + b has a fixed point F at x = b/(1 - a).

Definition. If f(x) =ax + b has a fixed point F then can be expressed in Fixed Point Form as f(x) =a(x - F) + F.