Iterating functions

Suppose you have a function f(x). Iterating it means you pick a value, plug that value into the function, get a value, take the new value and plug it into the function, and then get a second value.
You then take that value, plug it into the function and get a third value. You keep repeating this process and then you try to figure out what is happening to the values you are generating.

For example, take f(x)=x+1. If you start at 0 and keep evaluating the function and then evaluate the result you get an increasing sequence. Plug in 0 to start and you get the positive integers, 0,1,2,3 and the result just keeps getting bigger and bigger. What happens if you try f(x)=x-1? What about 2x+3, 5x-7, -8x-3? You get the idea.

If you choose a quadratic function things get more interesting. Look at the simplest quadratic If you plug in 0 what happens? What about 1 and -1? What about other values? Try both big and small values, both positive and negative. Can you think for general rules that would tell you how any number will behave?

Now let’s try some slightly more complicated quadratics. First try . Try the usual suspects 0,1 and -1. What is going on here? What about big and small numbers in this case?

Now try . How does it work?

Next steps:

Do you think you can say what will happen if you use a general quadratic? . (Hint: complete the square)

f(x)=
Starting Value:
Number of iterations:
Iterate
Results