Question

The change in water level of a lake is modeled by a polynomial function, W(x). Describe how to find the x-intercepts of W(x) and how to construct a rough graph of W(x) so that the Parks Department can predict when there will be no change in the water level. You may create a sample polynomial of degree 3 or higher to use in your explanations.

Answer 1

If x is time and W(x) is the change in water level at a certain time, then W(x) = 0 indicates when the water level does not change. In other words, the change in water level is 0.

This occurs exactly at the x intercepts as the x-intercepts are points of the form (x,0) where x is some number and the y coordinate is always 0. These special points are also known as roots. The roots or x intercepts are places where the curve crosses the x axis. The handy thing about roots is that they are visually easy to find, and relatively easy to comprehend no matter what math level you deal with. This is why many people of different backgrounds can understand what is going on even if they haven't taken a formal math course (in a while). So if you're giving a presentation, you can simply point to where the roots are and the managers would most likely understand. 

In terms of algebra, it depends on the complexity of the polynomial. For cubics and higher, you'll most likely need a graphing calculator or special software to get the approximate solution. Factoring and using the rational root theorem is a bad idea as it would take a while. It might not even be possible if the roots aren't whole numbers. Thankfully software makes the process relatively painless.