Solving Equations Graphically

So, in class today, some of you may have been wondering about the computer program that I was using. This utility is called Grapher and it is installed in any recent Mac OS computer. You'll find it in the Utilities folder within Applications.

I was wondering then whether there is a similar resource available for Windows computers as well. Doing a Google search on "Graphing Calculator" I found the following possibility: GraphCalc. I don't have immediate access to check this out, so I'd certainly welcome some comments here as to how well it works.

Now that you have something to work with (and a calculator will work as well, just a little slower), here is something interesting to notice. To solve ax=sin(x), we need to plot y=ax and y=sin(x). You also need to choose a value of a. In Grapher, you would add a New Equation (Cmd-Opt-N) like a=0.25. You now need to find where these graphs intersect.

In class, we learned that we can create new equations that have the same solutions by performing the same operation to both sides (other than division, where we worry about division by zero). So we could get a new equation like a=sin(x)/x. Now we plot y=a and y=sin(x)/x. If you add these as two new graphs instead of getting rid of the old plots, you can compare the two equations graphically. Here is the plot:
I used different colors to distinguish which intersections I was looking for.

For the original equation (ax=sin(x), shown in red), we see there are three intersections. For the new equation (a=sin(x)/x, shown in green), we only have two intersections. But those two intersections agree exactly with the original (see the highlighted intersection at the same value of x marked by circles), and the third corresponds to x=0 which disappeared because we divided both sides by x.