# Math Insight

### Applet: Function iteration using initial bacteria growth model

To test how well the model $B_{t+1}=\frac{5}{3}B_t$ with $B_0=0.022$ matches experimental data for the bacteria population, we can iterate the function $f(x)=\frac{5}{3}x$ with initial conditions $x_0=0.022$. For comparison, the bacteria data are plotted with purple diamonds. Click iterate a few times to see the solution $B_{t} = \left(\frac{5}{3}\right)^t 0.022$, shown in the right panel under the x_n heading and plotted as green dots in the right panel.

The effect of repeatedly applying the function $f(x)$ to the starting value $x_0$ is shown both by the list at the right and the graph at the left. At first, only the initial value $x_0$ is shown in the list and just the point $(0,x_0)$ is shown on the graph. (In the list heading, $x_n$ is written as x_n.) Each time you click the “iterate” button, the function is iterated by applying $f$ to the previous value, using the recursion $x_n = f(x_{n-1})$. Then, the new iterate $x_n$ appears in the list and the new point $(n,x_n)$ appears on the graph. Note that the iteration number $n$ is plotted on the horizontal axis (what you may normally think of as the $x$-axis), and the values of $x_{n}$ are plotted on the vertical axis (what you may normally think of as the $y$-axis). The values of each $x_{n}$ are also marked with horizontal lines and the last value $x_{n}$ is labeled. You can change the function $f(x)$ by typing a new function in the box. You can change the initial point $x_0$ by typing a new value in the box or dragging the blue point. You can zoom the vertical axis with the + and - buttons and pan up and down with the buttons labeled by arrows.