I realized recently that you can do something really cool with good students after they learn the standard forms for conic sections: you can compute the compactifications of their moduli spaces. I gave an undergraduate talk based on this, and I think it went really well. You have to wave your hands a bit and you might not want to use the word compactification. It is pretty obvious how to draw the uncompactified spaces of conic sections centered at the origins, but something really cool happens when you approach the boundary. I think this could be stretched out a bit longer than an hour and you could probably do several nice lectures on it, one for each different moduli space.
Let me know if you come up with any new low level examples for the moduli spaces. Mine were: -Triangles in the plane, -circles centered at the origin, -circles in the plane, -ellipses centered at the origin, -hyperbolas centered at the origin.
You could also look at how the discriminant is a function on the moduli space.