Iterating a Function in Desmos

Desmos won’t let you type a = f(a) and have it update itself — that’s an equation to solve, not a repeatable step. Instead, define the sequence as a function of the step number, the same way you’d write it in code.

Desmos set up to iterate cos(x) starting from 0.5

Steps

  1. Define the function you want to iterate:

    g(x) = cos(x)

  2. Define the starting value as step 0:

    a(0) = .5

  3. Define each step in terms of the one before it:

    a(n) = g(a(n-1))

  4. Make a list of step numbers:

    N = [0, ..., 100]

  5. Plot the points:

    (N, a(N))

  6. Click on the wrench and set the scale (approximately these values):

    0 <= x <= 100 -5 <= y <= 5

Desmos fills in the whole sequence automatically — no clicking, no typing it out by hand. If the points settle onto a flat line as N grows, that height is the fixed point; if they fly off the screen, it’s repelling.

Swap in a different g(x) or a different starting value in a(0) to try any of the functions from the Fixed Points handout.