Visual introduction to parabolas

In this video, we are going to talk about one of the most common types of curves you will see in mathematics, and that is the parabola. The word "parabola" sounds quite fancy, but we'll see it's describing something that is fairly straightforward.

Now, in terms of why it is called the parabola, I've seen multiple explanations for it. It comes from Greek, "parah," that root word similar to "parable." You could view it as something beside, alongside something in parallel. "Bola" has the same root as when we're talking about ballistics—throwing something. So you could interpret it as beside, alongside something that is being thrown.

Now, how does that relate to curves like this? Well, my brain immediately imagines, well, this is the trajectory; this is the path that is a pretty good approximation for the path of things that are actually thrown. When you study physics, you will see the path; you will approximate the paths of objects being thrown as parabolas, so maybe that's where it comes from. But there are other potential explanations for why it is actually called the parabola; it has been lost to history.

But what exactly is a parabola? In future videos, we're going to describe it a little bit more algebraically. In this one, we just want to get a sense for what parabolas look like and introduce ourselves to some terminology around a parabola.

So, these three curves, they are all hand-drawn versions of a parabola. You immediately notice some interesting things about them. Some of them are open upwards, like this yellow one and this pink one, and some of them are open downwards. You will hear people say things like "open up," "open down," "open downwards," or "open upwards." So it's good to know what they're talking about, and it's hopefully fairly self-explanatory.

Open upwards, the parabola is open towards the top of our graph paper here; it's open towards the bottom of our graph paper. This looks like a right-side-up U; this looks like an upside-down U right over here. This pink one would be open upwards.

Now, another term that you'll see associated with a parabola—and, once again, in the future, we'll learn how to calculate these things and find them precisely—is the vertex. You should view the vertex as the maximum or minimum point on a parabola. If a parabola opens upwards, like these two on the right, the vertex is the minimum point.

The vertex is the minimum point right over there, and so if someone said, "What is the vertex of this yellow parabola?" Well, it looks like the x-coordinate is 3 and 2, so it is 3 and 2. It looks like the y-coordinate is about -3 and 1/2. Once again, once we start representing these things with equations, we'll have techniques for calculating them more precisely. But the vertex of this other upward-opening parabola is the minimum point; it is the low point.

There is no maximum point on an upward-opening parabola; it just keeps increasing as x gets larger in the positive or the negative direction. Now, if your parabola opens downward, then your vertex is going to be your maximum point.

Now, related to the idea of a vertex is the idea of an axis of symmetry. In general, when we're talking about—well, not just three dimensions, but even two dimensions—but especially in two dimensions, you can imagine a line over which you can flip the graph. You can flip the graph, and so it meets; it folds onto itself.

The axis of symmetry for this yellow graph right over here, for this yellow parabola, would be this line. Actually, draw it a little bit better; it would be that line right over there. You could fold the parabola over that line, and it would meet itself.

That line—I didn't draw it as neat as I should—that should go directly through the vertex. To describe that line, you would say that line is x = 3.5. Similarly, the axis of symmetry for this pink parabola should go through the line x = -1.

So let me do that. That's the axis of symmetry; it goes through the vertex, and if you were to fold the parabola over it, it would meet itself. The axis of symmetry for this green one should once again go through the vertex. It looks like it is x = -6. Let me write that down; that is the axis of symmetry.

Now another concept that isn't unique to parabolas, but we'll talk a lot about it in the context of parabolas, are intercepts. So when people say "y-intercept," and you saw this when you first graph lines, they're saying where does the curve intercept or intersect the y-axis.

So the y-intercept of this yellow line would be right there; it looks like it's the point (0, 3). The y-intercept for the pink one is right over there. At least on this graph paper, we don't see the y-intercept, but it eventually will intersect the y-axis; it just will be way off of this screen.

Now, you might also be familiar with the term "x-intercept," and that's especially interesting with parabolas, as we'll see in the future. The x-intercept is where you intercept or intersect the x-axis. Here, this yellow one does so in two places, and this is where it gets interesting.

Lines will only intersect the x-axis at most once, but here we see that a parabola can intersect the x-axis twice because it curves back around to intersect it again. So, for here, the x-intercepts are going to be the points (1, 0) and (6, 0).

Now you might already notice something interesting: the x-intercepts are symmetric around the axis of symmetry. They should be equidistant from that axis of symmetry, and you can see they indeed are. They're both exactly 2.5 away from that axis of symmetry.

If you know where the intercepts are, you could say the midpoint of the x-coordinates would give you the axis of symmetry—the x-coordinate of the axis of symmetry and the x-coordinate of the actual vertex. Similarly, the x-intercepts here look like they're the points (-7, 0) and (-5, 0), and the x-coordinate of the vertex or the line of symmetry is right in between those two points.

Now, it's worth noting not every parabola is going to intersect the x-axis. Notice this pink upward-opening parabola; its low point is above the x-axis, so it's never going to intersect the actual x-axis. So, this is actually not going to have any x-intercepts.

So, I'll leave you there; those are actually the core ideas or the core visual themes around parabolas, and we're going to discuss them in a lot more detail when we represent them with equations. As you'll see, these equations are going to involve second-degree terms, so the most simple parabola is going to be y = x^2.

But then you can complicate it a little bit; you could have things like y = 2x^2 - 5x + 7. These types—and we'll talk about, in more general terms, these types of equations, sometimes called quadratics—they are represented generally by parabolas.