Math Insight

Applet: 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.

Applet file: function_iteration_cobweb.ggb

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