Perfecting Our Perception Of Parabolas

Parabolas! They’re great! They’re our friends! Our best friends! We know so much about them! But there are some things about them we don’t yet know. And even for the things we do know about parabolas, we only know them for the handful of specific parabolas we’ve played with on the problem sets.

We want to know more things about more parabolas.

We want to know everything about every parabola.

Specifically: let’s find the roots not just of some parabolas, but of every parabola. Let’s find the vertex not just of some parabolas, but of every parabola. And, along the way, let’s discover this deep truth: the vertex of every parabola is exactly halfway betwen its roots.

OK, full disclosure, I know you know this stuff already. You know the theorem about how the vertex is halfway in between the roots. You know the formula for the roots of every parabola. That’s just the quadratic formula.

Or… you “know” this stuff. You’ve seen it. You don’t really know it. Not yet. You’ve been told it. You’ve read about it. Ought you believe everything you read? You’ve trusted your teachers, but you shouldn’t, because we’re human beings, and we’re mortal, and we’re falliable. We’re unreliable narrators. You should never trust your teachers—you should only trust the mathematics! It’s not your teachers who are in charge—it’s mathematics. We’re just as subservient to its universal laws as you are. There’s no “because I say so” in mathematics.

That’s the approach I want us to take here. I want us to trust, and feel, and hear, and listen to the mathematics, ourselves. I want us to build and create and find and discover this stuff—the quadratic equation, this fun theorem—for ourselves!

The fancy math word for “discover” is “prove,” but I’m avoiding that word, ’cause it has a lot of baggage. Proofs are supposed to be hard and scary! And sometimes they are. But not these! These proofs are the same things we’ve been doing with parabolas for weeks now—except that instead of using a specific number, we’re using some random letter. That random letter can stand for any number, and so by doing these calculations with a letter instead of a number, we’ll effectively be solving them for every number. (Infinity is ours!)

So:

• Consider the parabola:  \[f(x) = (x-r)(x-s)\]
  • Sketch this. Where are its roots? Where is its \(y\)-intercept?
  • Then, use the forcing-it-into-vertex-form method (a.k.a. “completing the square”) to find the vertex! Find its \(x\) and \(y\)-coordinates. Do you notice anything interesting about the relationships between \(r\), \(s\), and the vertex?

• Consider the parabola: \[f(x) = ax^2+bx+c\] For this problem, I want you to forget all about the quadratic equation. You have no idea what it is, or even that it exists. After all, you don’t know why it’s true and how it works—you only know it because some teacher told you it was true and you believed them. But that’s a bad idea: you should never trust your teachers! You should only trust the mathematics!

  Anyway, we’d like to draw a picture of this parabola. But, in its current form, we can’t say much about it, other than that it has a \(y\)-intercept of \(c\). So let’s try to find the roots and the vertex.
  • Force this parabola into vertex form. What are the \(x\) and \(y\)-coordinates of the vertex?
  • Then, now that you’ve got it in vertex form, find where the roots are!
  (Of course draw a picture, too!)

Meanwhile…

You’ve read a lot of my writing this year. It’s time to turn the tables. I want to read some of your writing.

Write up these problems. These are both remarkable results, and they deserve care commensurate with that! Explain these two problems and how you get the answers, typed. Show all of your steps and explain your entire thought process, using both mathematical symbols, beautiful English prose (i..e, full sentences and paragraphs), and pictures. The goal is to explain, as if you were explaining how to solve these two problems to a friend in a letter (clearly, engagingly, lucidly). (Of course include a heading, your name, the date, etc.)

Make it nice. Write something you’re proud of.

(Oh, and postscript: please don’t use ChatGPT or another LLM! I love LLMs; I use them every day; I think they’re amazing, but with this, I want to read you. It’s like how I don’t want you to use Desmos to make graphs: it’s because the point isn’t getting the graph so much as it is the process and the understanding. With this assignment, I really do want to read what you write.)