Extending graph control

This article takes my idea from yesterday’s post about using $g(x)=\sqrt \frac{\left | x \right |}{x}$ to control the appearance of a graph and extends it in two ways.

• Part I below uses Desmos to graph $y=(x+2)^3x^2(x-1)$ from the left and right simultaneously
• Part II was inspired by my Twitter colleague John Burk who asked if this control could be extended in a different direction.

Part I: Simultaneous Control

When graphing polynomials like $y=(x+2)^3x^2(x-1)$, I encourage my students to use both its local behavior (cubic root at $x=-2$, quadratic root at $x=0$, and linear root at $x=1$) and its end behavior (6th degree polynomial with a positive lead coefficient means $y\rightarrow +\infty$ as $x\rightarrow\pm\infty$). To start graphing, I suggest students plot points on the x-intercepts and then sketch arrows to indicate the end behavior.  In the past, this was something we did on paper, but couldn’t get technology to replicate it live–until this idea.

In class last week, I used a minor extension of yesterday’s idea to control a graph’s appearance from the left and right simultaneously.  Yesterday’s post suggested  multiplying  by $\sqrt \frac{\left | a-x \right |}{a-x}$ to show the graph of a function from the left for $x.  Creating a second graph multiplied by $\sqrt \frac{\left | x-b \right |}{x-b}$ gives a graph of your function from the right for $b.  The following images show the polynomial’s graph developing in a few stages.  You can access the Desmos file here.

First graph the end behavior (pull the a and b sliders in a bit to see just the ends of the graph) and plot points at the x-intercepts. From here, you could graph left-to-right or right-to-left.  I’ll come in from the right to show the new right side controller. The root at $x=1$ is linear, so decreasing the b slider to just below 1 shows this. Continuing from the right, the next root is a bounce at $x=0$, as shown by decreasing the b slider below 0.  Notice that this forces a relative minimum for some $0. Just because it’s possible, I’ll now show the cubic intercept at $x=2$ by increasing the a slider above 2. All that remains is to connect the two sides of the graph, creating one more relative minimum in $-2. The same level of presentation control can be had for any function’s graph.

Part II: Vertical Control

I hadn’t thought to extend this any further until my colleague asked if a graph could be controlled up and down instead of left and right.  My guess is that the idea hadn’t occurred to me because I typically think about controlling a function through its domain.  Even so, a couple minor adjustments accomplished it.  Click here to see a vertical control of the graph of $y=x^3$ from above and below. Enjoy.