High School: Geometry
Expressing Geometric Properties with Equations HSG-GPE.A.2
2. Derive the equation of a parabola given a focus and directrix.
What do your satellite dish, the path that Katniss Everdeen's arrow would follow if she aimed up in the sky, and the water swirling around inside a flushing toilet all have in common? Parabolas. Yes, you heard us right. The shape you get from slicing a cone perpendicularly to its base shows up in plenty of unusual real-world places.
Students should already know that parabolas, like pretty much everything else in the Mathland, can be labeled with their own unique equations in the form y = ax2 + bx + c. But giving your students the values of a, b, and c would just be too easy, don't you think? Instead, have them find the equation when given a focus and directrix.
Students should know that a focus is a point near a parabola and a directrix is a line near a parabola. So what's so special about them? In Mathese, the parabola is the set of all the points that are the same distance between two things: a given point and a given line. In picture form, that would be something like this.
Students should know that because the distances are equal, they find the equation of a parabola by applying the bulky equation below, where xf and yf are the coordinates of the focus and either xd or yd can be substituted with the value of the directrix. The other value becomes its respective variable (so either xd becomes x, or yd becomes y).
To help students become more efficient, tell them to square both sides and then expand them. Then, they can solve for y to get the equation of the parabola. No sweat (unless they happen to be on a treadmill or something).
Students can also use the (x – h)2 = 4p(y – k) formula for a horizontal directrix and (y – k)2 = 4p(x – h) for a vertical directrix. They should know that (h, k) are the coordinates of the parabola's vertex and p is the distance from the focus to the vertex. Remind students that p will be negative when the focus is either under or to the left of the directrix.
Sideways parabolas may exist in the real world when the satellite dish gets knocked over or when you stand on a platform and throw a boomerang. Hopefully, the watery flushing toilet parabola never goes sideways.
That would be gross.