Presentation on Parabola

Question

All right, I'm presenting a class in front of some $20$ or so $12^\text{th}$ grade students in a couple of days on the topic of Parabola. I was wondering what would be the best way to teach it to them, and also if anyone had some really amazing problems as I have to give them $2$ insanely hard problems\ or just problems with a really great solution or analysis to be done in the class. Also if anyone had their hands on any great lecture notes from college I would be very grateful since we have been instructed to not restrict our self to any sort of level of difficulty.
Any ideas would be welcome.

Answer 1

Too long for a comment ... One special Archimedean Lemma I've used for a lesson: Let $f\colon x\mapsto ax^2$ for positive $a$ and let $P(p_1,p_2)$ a point with $p_2<0$. The two tangents from $P$ to $f$ touch $f$ in $A_1$ and $A_2$, resp., midpoint of $A_1A_2$ is $M$. Now show that the midpoint $S$ of $MP$ is a point of $f$.

Use this fact to cleverly construct the tangent in any point $A$ of $f$.

Answer 2

I never taught this subject, but if I were to I would start from the definition as section of a circular cone,
and their projective properties, invariants, and pass after to the metric properties. So I would start with
- how is it projected by the cone on a plane normal / parallel to its axis;
- shifting the sectioning plane, what the resulting parabolas and their projections have in common; - same, keeping instead firm the sectioning plane and widening the angle of the cone; - consider the normals to the cone in the plane of the parabola, where do they converge in the projection onto a plane normal to the cone axis, etc. - considering the cone double, and two symmetric parabolas on the two foils, where do their planes cross, what is the meaning of that line;
etc.

Concerning an interesting exercise, I would propose that of reproducing Archimedes steps in computing the area of a parabolic segment, centuries before calculus.