Precalculus students already have a ton of background information about parabolas, so it can be difficult for them to switch gears and think about parabolas in terms of a focus and directrix. In this video I discuss why I think these are the most difficult conic sections and how to support student mastery by using all of the parabola tools they know.
Like our work with ellipses and hyperbolas, we are going to start today by defining a parabola in three different ways. I give students a few minutes to define a parabola as a locus, a cross section, and an algebraic equation.
My students usually remember a little about the locus definition, but not all of it. I will ask them for associated vocabulary (focus, directrix, etc.) and we can usually piece it together as a class. For the algebraic equation, most students only give me the ones they know from Algebra (y = a(x – h)2 + k and y = ax2 + bx + c).
It is good that students know different forms of equations of a parabola, but neither of these gives us information about a focus or a directrix. I instruct students that we need a new parabola tool that will take these important aspects of the parabola into account.
I haven’t derived any of the conic section equations because students have seen them in Algebra 2. Because the parabola equation is so foreign to what they know already, I think it is important to take the time to show (or review) where it comes from. It is also a very simple derivation and really builds the conceptual understanding of the focus and the directrix.
Here is the work from the derivation that we did in class. As you can see, I started by choosing a few points on the parabola and drawing in distances to the focus and directrix to show that they are congruent. Then we used the distance formula to derive our equation. Since our parabola was centered at the origin, we simply thought of the transformations if the vertex was at (h, k).
Finally, we discuss how the p value (the distance from vertex to focus) is a directed distance and the sign of the number will tell us which way the parabola opens.
Finally I give an assignment from our textbook to recap all of the information about parabolas. I try to incorporate many different types of problems so that students are exposed to finding the equation given the focus and directrix or given the vertex and a point on the parabola, for example.