Topic A: Lessons 8-10: Parabolas intro
- [Instructor] 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 para, that root word, similar to parable. You could view of something beside, alongside, something in parallel. Bola, 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 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'll 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 gonna describe it a little bit more algebraically. In this one, we just wanna get a sense for what parabolas look like and introduce ourselves to some terminology around a parabola. These three curves, they are all hand-drawn versions of a parabola, and so you immediately notice some interesting things about them. Some of them are opened upwards like this yellow one and this pink one, and some of them are open downwards. You will hear people say things like open, opened down, open downwards or open down or open upwards, so it's good to know what they are 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 the parabola, and once again, in the future, we'll learn how to calculate these things and find them precisely, is the vertex. The vertex you should view as the maximum or minimum point on a parabola. So 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, looks like the x coordinate is three and a half, so it is three and a half. It looks like the y coordinate, it looks like it is about negative three and a half. 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, it 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, two dimensions but even three dimensions, but especially in two dimensions, you can imagine a line over which 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, it would be this line. I'm gonna have to 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. And that line, I didn't draw it as neat as I should, that should go directly through the vertex, so to describe that line you'd say that line is x is equal to 3.5. Similarly the axis of symmetry for this pink parabola, it should go through the line x equals negative one, 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? It should, once again, go through the vertex. It looks like it is x is equal to negative six. This is, 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 graphed lines, they're saying where is the graph, 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 zero comma three, zero comma three. 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. You might also be familiar with the term x-intercept, and that's especially interesting with parabolas as we'll see in the future. X-intercept is where do you intercept or intersect the x-axis? Here this yellow one you see it does it two places, and this is where it gets interesting. Lines will only intersect the x-axis once at most, but here we see that a parabola can intersect the x-axis twice, because it curves back around to intersect it again, and so for here the x-intercepts are going to be the point one comma zero and six comma zero. You might already notice something interesting. The x-intercepts are symmetric around the axis of symmetry, so they should be equal distant from that axis of symmetry, and you can see they indeed are. They are both exactly two and a half away from that axis of symmetry, and so if you know where the intercepts are, you just take, you could say, the midpoint of the x coordinates, and then you're going to have the axis of symmetry, the x coordinate of the axis of symmetry and the x coordinate of the actual vertex. Similarly the x-intercept here looks like it's negative, the points are negative seven comma zero and negative five comma zero, and the x coordinate of the vertex, or the line of symmetry, is right in between those two points. It's worth noting not every parabola is going to intersect the x-axis. Notice this pink upward-opening parabola, it's 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. 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 is equal to x squared, but then you can complicate it a little bit. You could have things like y is equal to two x squared minus five x plus seven. These types that we'll talk about in more general terms, these types of equations sometimes called quadratics, they are represented, generally, by parabolas.