# Common Core Standards: Math See All Teacher Resources

#### The Standards

# 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* = *ax*^{2} + *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 *x _{f}* and

*y*are the coordinates of the focus and either

_{f}*x*or

_{d}*y*can be substituted with the value of the directrix. The other value becomes its respective variable (so either

_{d}*x*becomes

_{d}*x*, or

*y*becomes

_{d}*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} = 4*p*(*y* – *k*) formula for a horizontal directrix and (*y* – *k*)^{2} = 4*p*(*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.