** Using WeBWorK **

I was able to solve the problem. It seems that the file geogebra.jar is not present on the WeBWorK server at my university. This file appears in the Applet Setup part of the pg file around here:

$applet = JavaApplet(
code => "geogebra.GeoGebraApplet",
archive => "geogebra.jar",
codebase => findAppletCodebase("geogebra.jar"),
...

I was able to find the geogebra.jar on the geogebra website at

http://www.geogebra.org/webstart/4.2/unpacked/geogebra.jar

and changed the pg file:

code => "geogebra.GeoGebraApplet",
archive => "http://www.geogebra.org/webstart/4.2/unpacked/geogebra.jar",
codebase => findAppletCodebase("http://www.geogebra.org/webstart/4.2/unpacked/geogebra.jar"),

The 4.2 in the URL refers to the version of GeoGebra you use, so you may have to change it.

Meanwhile, I also solved the issue with my applet not graphing all vertices. I managed to get it to plot all vertices and edges in my graphs now. I'll try some other techniques like getting answers from the applet next week.