Orbits

1. Orbits of $f(x) = x^2$.
   
   1. Complete the orbits for the function $f(x) = x^2$.
      

      $x_0$	$.5$	$2$	0	$-3$	$-.4$	x
      $f(x_0)$
      $f^2(x_0)$
      $f^3(x_0)$
      $f^4(x_0)$
      $f^5(x_0)$

      
      

   2. If you can, write down a formula for the $n^{{\rm th}}$ iterate of $f$, $f^n(x)$.
      
      
      
      
      
      
      
      
      
   3. Summarize your findings. Describe the dynamics of $f(x)$. Are there any fixed points? Are there any cycles? Are any points eventually fixed or eventually cyclic? What happens to different initial conditions?

2. Orbits of $g(x) = \frac{1}{2}x + 2$.
   
   1. Complete the orbits for the function $g(x) = \frac{1}{2}x + 2$.
      

      $x_0$	$.5$	$2$	0	$-3$	$-.4$	x
      $g(x_0)$
      $g^2(x_0)$
      $g^3(x_0)$
      $g^4(x_0)$
      $g^5(x_0)$

      
      

   2. If you can, write down a formula for the $n^{{\rm th}}$ iterate of $g$, $g^n(x)$.
      
      
      
      
      
      
      
      
      
   3. Summarize your findings. Describe the dynamics of $g(x)$. Are there any fixed points? Are there any cycles? Are any points eventually fixed or eventually cyclic? What happens to different initial conditions?
      

   4. Sketch a time series graph for $g(x)$ for a few different seeds.