I have recently completed most of the topics related to first derivative test, concavity, curve sketching....
I still need to go back and review and re-teach some of this material (I find as I grade papers).
One thing I like to hammer home in my course, is the danger of over-reliance on technology, and that computers/calculators are a useful supplementary tool to analytical work, but not a replacement.
To that end, I would like to do more work with graphing functions with "hidden behavior", where graphing them in the standard viewing windows makes it appear that there are not so many max's or min's as there truly are, and one would likely not choose a correct viewing window without analytical work to give you a clue _where_ the extrema, etc., are likely to lie.
At an AP weekend conference in Phx, AZ I believe it was Richard Jensen who showed how to easily compose, for instance, cubic functions with such properties. I wish I could recall the method, or find my notes. <sigh>
Can anyone give me some tips on how to easily come up with such functions? I would like it, if possible, to be able to do non-polynomial functions as well.