Math Insight

Function iteration. 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.

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Visualizing function iteration via cobwebbing. Cobwebbing is a graphical method of exploring the behavior of repeatedly applying a function $f(x)$ beginning at an initial point $x_0$, shown as the blue point on the $x$-axis. Click the button labeled step to apply $f$ to $x_0$, obtaining $x_1=f(x_0)$, as shown in the list at the right and on the $y$-axis. Graphically, to get from $x_0$ to $x_1$, move vertically to the graph of $f$ (the blue curve). and then horizontally to the $y$-axis, as shown by the green line segments. Click step again to visualize how to find where the value $x_1$ is on the $x$-axis. As shown by the green line segments, one can reflect $x_1$ to the $x$-axis by moving horizontally to the line $x=y$ (red line), then vertically to the $x$-axis. To calculate more $x_i$ from the recursion $x_i=f(x_{i-1})$, repeat the process by clicking step two times for each iteration. To speed the process and produce a more traditional cobweb plot, clear the checkmark labeled “details.” In this case, the step button changes to an iterate button, which executes both steps with one click. In this case, the line segments are not extended to the coordinate axes but show the shortcut method of moving vertically to the graph of $f(x)$ (blue curve) and horizontally to the line $y=x$ (red line). 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 in and out with the + and - buttons as well as pan in different directions with the buttons labeled by arrows.

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Visualizing function iteration via cobwebbing, combined with plot of solution. Cobwebbing is a graphical method of exploring the behavior of repeatedly applying a function $f(x)$ to an initial value $x_0$. The left panel shows a cobweb plot while the right panel shows a plot of the results versus iteration number. The initial value $x_0$ is shown as the blue point on the horizontal-axis in the left panel and the blue point on the vertical axis in the right panel. At first just $x_0$ is shown. Each time you click the “iterate” button, the function is iterated by applying $f$ to the previous value, using the recursion $x_n = f(x_{n-1})$. The cobweb plot at the left shows how the new value of $x$ can be obtained by moving straight up or down from the previous point to the graph of $f$ (the blue curve). This process gives the new value $x_n$ as the vertical coordinate of the point where one hits the graph. To translate that value of $x_n$ to the horizontal coordinate, one moves left or right to the diagonal line (the red line). Then, one can find the next value by moving up or down to the graph of $f$ again. The new values are simultaneously plotted in the right panel as the points $(n,x_n)$. 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 one of the blue points. You can zoom in and out with the + and - buttons as well as pan in different directions with the buttons labeled by arrows.

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