Oh snap, parabolas!

            Greetings all, I just wanted to take a moment to discuss one of my favorite mathematical concepts: the parabola.  Why get so excited over those little u-shaped lines, you ask?  Because parabolas are awesome, that’s why.  This is due in no small part to the obscene amount of applications they have in real life.  In fact, when translated, its name literally means “application.”  Behold:

Calculating flight trajectory?  Parabolas

Constructing radio and satellite dishes so we can watch Netflix?  Parabolas

Automobile headlights?  Parabolas

Arched bridges so said automobiles don’t plummet into the water?  Parabolas

So what exactly is a parabola then?  It’s basically a two dimensional U-shaped curve which is vertically symmetrical about its vertex.  To break things down nice and simple, just imagine this: take the uppercase letter “U”.  Now draw a vertical line directly across its center.  See how both sides are split evenly?  That is vertical symmetry.  The very bottom point on the letter “U” is the vertex.  This referred to as its minimum, because there is nothing else below it.  If we were to rotate the parabola 180 degrees so that the “U” was upside down, the vertex is then referred to as its maximum, because there is nothing “higher” than that point.

We can define parabolas mathematically by using the standard form quadratic function, which is this: y = ax^2 + bx + c, where a≠ 0.  People tend to freak out when first introduced to this function because it looks scary, but it’s really not so bad.  Let’s take a look.

The first part of the quadratic function is that “ax^2” bit.  The “a” variable is called the leading coefficient and it determines how wide or narrow the U shape is.  If “a” is greater than or equal to 1, then the parabola is going to look skinny because it grows at a high rate.  The higher the number, the skinnier it gets.  And of course, when “a” is negative, nothing changes, it just means the parabola is upside down now.  If “a” is less than 1 but greater than 0 (example: ½) then the parabola is going to look stretched.  Remember those graphs of exponential functions, such as ab^x, which looked like this:

Kind of looks like a parabola cut in half at its vertex, right?  Recall that when we increased base “b” the steeper the growth was.  That’s similar in concept to what’s going on with "ax^2" in the parabola.

  We can do all sorts of cool things, like finding the vertex of a parabola by using the expression –b/2a.  All we have to do is look at ax^2 + bx + c, and plug in the corresponding values of “b” and “a”.  So for example, if I was a pilot in a reduced gravity aircraft and I wanted to calculate when to climb and when to level off at the vertex of my flight path, I’d need to use the quadratic function.

In another example, if I was shooting a cannonball and I wanted to know where it would land, I would have to use the quadratic function again.  In this case, I would also have to know where the x-intercepts were in the function.  I do this simply by setting y=0 and solving for “x” (example: 0 = ax^2 + bx + c).  When we look at the graph, we could then pretend the x-axis was the earth surface, the parabola is the arc of the cannonball’s flight, and the x-intercepts were the launching point and impact point.

See, parabolas are pretty cool stuff.  We wouldn’t be able to do a whole lot without them.  Cheers!