# Parabolas: Focus and Directrix

I am teaching conics now. I usually skip teaching anything about parabolas in depth because… well, they do so much with quadratics in Algebra II… and I would rather devote my time to something new. However this year I’m teaching with another teacher who did cover parabolas. So I had to learn what a focus and directrix is. I mean, I knew ages ago, but who needs to keep that kind of information in your head?!

For those who aren’t in the know, for me the big idea is that we can conceptualize a parabola as the result of graphing the algebraic equation $y=ax^2+bx+c$. But there is a second way to concieve of the same mathematical object: with a geometric argument.

If you have a piece of paper with single point drawn, and a single line (that doesn’t contain the point) drawn, those two objects uniquely define a parabola.

That’s a pretty awesome thing, once I started thinking about it. An alternative way to view something that I only ever think about in the standard “graph a quadratic” way!

## The Forwards Question

So given a point and a line, how can we draw this parabola? Here is how…

The point is the blue X. The line is the black line. We want to drag the red point along this vertical line so that the distance from the blue point to the red point is equal to the distance between the red point and the black line. So we use a ruler, some trial and error, and find that red point belongs somewhere here… [1]

And then we leave that red dot there, and start again with another vertical line. And find another point on that vertical line which has the same property!

And again and again and again. Until you have created a whole bunch of red points. Those form a parabola.

I’m still not 100% sure how I’m going to introduce this notion to my kids. I’m pretty sure I’m going to give each kid a printed paper that looks like

And ask them where to place the red dot… And then see if they can find a more efficient way than using a ruler and guessing a checking. (Paper fold! See it? If not, read the footnote.) I will probably do this as a warmup one day — and then have kids go “whaaaaat is this for?” and I’ll shrug and say “Wish I knew, kids…” and then move on not referencing this.

And then the next day for the warmup, I’ll find a way to have the whole class collect points for the same blue point and black line… We’ll generate the locus of all these points which are equidistant from the blue point and perpendicular distance to the black line… and lo and behold… the parabola. And then we’ll do the patty paper folding thing down in the footnote video.

So… Yeah. Now we have an obvious place to go…

## The Backwards Question

Here it is: Given a parabola, can you find the defining point and line? (The fancy mathematical words for these defining objects are “the focus” and “the directrix.”)

And so I created a sheet to have my kids figure out how to find these objects given a parabola. [Note: I haven’t used the sheet. I haven’t even worked out the sheet and made a key. I just whipped it up now! So apologies for any errors, if any.]

2016-04-25 Parabolas [docx form]

(1) I give footnote 1.

(2) I give 3c. In fact, partly I think giving 3a is a bit much as is.

Both give away too much. So why didn’t I change it? Do I not have confidence in my kids?

No. It’s because I wasn’t even planning on introducing parabolas. And now I got sucked into them — learning all about them — and I am excited to share some of this stuff with my kids. But I don’t have the time for this. The fact that I’m going to give about a day for parabolas is more than I was planning… so I have to keep things a bit on the crisper side.

What else would I change if I had more time? I would have kids think about if this works for an “upsidedown” parabola. And also have them use what they know about inverse functions to apply this to “sideways” parabolas.

I honestly don’t know if I’m going to use this in class. I probably will because I took the time to make it, and I kinda got excited when I was figuring out for myself all this focus/directrix stuff. I pretty much took this definition of a parabola and figured all this out myself — and I hope kids get the same joy. But have I convinced myself that kids need to learn about a parabola other than there is this other way to “create” them that isn’t algebraic? Is there a “big idea” hidden in this worksheet? I don’t think so. This may be a one-time use worksheet.

[1] Now in actually, there is an easy geometric way to find that red point. It involves a simple paper fold. Fold the blue point to the point on the directrix below the red point. What that crease intersects the vertical line is where the red dot should be. Perpendicular bisectors FTW! And you can do a quick patty paper demonstration of this to create a parabola! (We did this in my class last year, for parabolas, hyperbolas, and ellipses, thanks to Tina C.)

1. I love the “all parabolas are similar” idea. If I had more time (maybe next year?) I will add that in.

1. ctrujillo says:

Look up hot dog cookers. You will now love the focus definition of parabolas. My students will be making them very soon.

1. Haha I always *show* my kids pictures of this when I introduce the idea of the parabola. But in the past, I just show them this definition — and how it can be applied in *the real world*, and then move on. No equations. I spent my time on hyperbolas and ellipses. But now, the hotdog cooker will have real meaning for them!!!

2. Sam, do you talk about parabolas that don’t have a horizontal directrix? Might be hard to write the equation without linear algebra, but conceptually makes sense with the geometric method.

1. Nope. It’s in the book we use, and we can use our sum of angles trig formulas to show how to do this… but no time. :(

3. Sam, I figured you’d like to see Mrs. G’s ideas behind showing the parabola.

http://givemeasine.blogspot.com/2016/03/whats-parabola-anyway.html

Since students already “know” what a parabola is, you can use scaffolding to show this “other” definition.

Though I tend to disagree with your assessment that a quadratic equation defines a parabola. Parabolas were originally conic sections and somebody eventually noticed that this conic section and the graph of a quadratic looked very similar. So giving the students the REAL definition of parabolas can solidify their understanding of quadratics a bit further.

1. Yes! This is great. I will do this on Monday in class!!! Thank you for sharing.

1. My pleasure! I did weep at its beauty. :)

4. Mr. K says:

TIL: The perpendicular bisector between the focus and any point on the directrix is tangent to the parabola.

Whooda thunk.

1. So awesome! I will share this with a student who likes building things!!! I wonder if he can build this!

1. Oh, that would be so great! If he does build it, I’d love to see a picture!

1. They have a giant sculpture illustrating this property at the Museum of Math in NYC! So cool!

5. Hey Sam, I was reading this post and I couldn’t help but reflect on how I taught this topic (and conics) a couple years ago during my student teaching. While it was far from stellar, I recall I centered everything with conics around the concept of distance. I started the unit with circles (using the concept of a set of points equidistant from a single point), then I proceeded to have students work to define parabolas using distance (a set of points equidistant from a point and line). Students worked with the distance formula/Pythagorean theorem a lot to derive the equation. Here’s a series of tasks I used to get students to derive the general equation for a parabola using distance: https://docs.google.com/docume
nt/d/1xy5KEYLVYKbkdbXDZFasA_Ev6DCZ3Qy_OIaSYw1c9gI/edit?usp=sharing

1. Thank you for sharing!!!