 One topic of great interest to both ancient geometry and even the modern approach of analytics geometry are the so-called conic sections. They get their names from the possible intersections one can make with a plane and a cone, or technically it's a double cone, and you can see the possibilities in front of you. The first possible conic section if you hit your cone with a plane is you actually form what we would usually call a parabola. All right, and college algebra one, we've talked a lot about parabolas, lots of different things we could say, and we have our standard to desert the vertex form of a parabola y equals a times x minus h squared plus k and of significance here was this point h comma k, which act as the vertex of said parabola here. Although we've seen previously that you could slightly re, you could rework this thing and get something of the following form for p times y minus k is equal to x minus h squared, and that way you could talk about this idea of the focus of the parabola for which you might think it's something like here, and the focus was significant because as particles enter the parabola, they always bump towards the focus or things leave the focus, they always bump out and come out straight. So this is nice reflective property for parabolas, and this is a consequence of it being one of these conic sections. The second conic section that we've talked about in this series is the ellipse, like you can see right here. The plane could intersect the cone forming an ellipse. Now the standard equation of an ellipse typically looks like the following. You get x minus h squared over a squared plus y minus k squared over b squared is equal to 1. Now I should mention with the parabola, the location of the square could be on either location, right? Because are we going to square the y-corner, are we going to square the x-corner? That makes a difference between do we have like a concave up parabola or a concave right parabola. With an ellipse, because of its symmetry, you don't have to worry about those things as at all. Now a special type of an ellipse is actually a circle. So some people consider this a fourth conic section. A circle, just the usual equation, x squared, sorry, x minus h squared plus y minus k squared is equal to r squared. The circle is just a special type of ellipse where both the parameters a equal b are the same. We often call it r in that situation. Now it turns out for ellipses, they also have sort of an idea of a focus, right? Unlike a parabola which has a single focus, an ellipse actually has two, two so-called foci, the plural there, which these foci have the property, a very similar reflective property to the parabola here. That is a particle emanates from one focus. It'll always bounce back towards the second focus. And so people have used this idea to create things called like whispering chambers. You're in a large elliptical room. If you have one focus, you can actually talk to someone as in a whisper at the other focus, even though you could be tens or hundreds of feet apart, right? Sort of case in point, if you look at the Latter-day Saint Tabernacle that's built in Salt Lake City, it was built in the 1800s by the Mormon pioneers at the time. And so this was before, of course, you know, and they're on the frontier before there was any kind of modern-day electricity, but they built this elliptical building so they have large religious services and such and what had the way they designed it is actually that the pulpit is at one of the foci of the ellipse and then the other foci is actually in the audience somewhere. If you go on a tour of the Tabernacle there, the tour guides can actually show you this is the other foci here. And so the thing is, you could hear the speaker as if they were right next to you because of the reflective property, the acoustic benefits of that. And so even if you're not sitting in that focal seat, if you're anywhere close to it, it definitely magnifies the volume because it feels like the person is talking to you just from a few feet away instead of like a few hundred feet away. And so there's there are some engineering benefits to the reflective property of the foci of an ellipse. So what I want to talk about now is actually the third and final type of conic section and this is what's known as the hyperbola. So if you tilt your plane in such a way, it turns out that the plane could intersect the both the top and bottom cone simultaneously and the intersection forms what we call a hyperbola. Now, the general equation for a hyperbola will look like one of the following. It'll look like x minus h squared over a squared minus y minus k squared over b squared equals 1. Or it could look like y minus k squared over b squared minus x minus h squared over a squared equals 1. In which case, unlike the ellipse, which has a plus between it, the hyperbola is formed by taking the same formula, but it's taking a negative sign. Now the negative could be in front of the x-coordinate or it could be in front of the y-coordinate. And it'll have different effects on the graph based upon what you see there. And so let's actually switch over to Desmos for a second and let me show you what you can get here. So if you were to take the equation x squared minus y squared equals 1, you get what we will call a horizontal hyperbola. So like I said, we'll call this a horizontal hyperbola. And a hyperbola kind of looks like you have two parabolas looking at each other. It's not exactly true, but in terms of concavity, a horizontal hyperbola will concave to the left and to the right. This is a distinction to the vertical hyperbola, which we'll see in just a second. But this is the basic graph of this hyperbola. And again, it's formed by having the plane intersect the cone at two different places. If we were to switch things up, if you actually put the y squared first, y squared minus x squared, you see now an example of a so-called vertical parabola, excuse me, vertical hyperbola. With the vertical hyperbola, you see going on here that it's going to concave both up and down simultaneously. And so that's what we mean by vertical hyperbola. So vertical or formed when you have a positive y and negative x, the horizontal is formed when you have a positive x squared and a negative y squared as well. And so this hyperbola, some things we should mention, I'm going to switch back to the horizontal one for this exercise here. Well, some important things to pay attention to are going to be the vertex. Just like a parabola has a vertex, just like a ellipse has a vertex, the hyperbolas also have vertices as well. And so we form these by horizontal shifts and vertical shifts. So if we have these parameters, I'm going to plot the point h comma k right here. You can see there in orange. So the vertex actually is not part of the hyperbola, just like the center, the center, I guess that's actually what you usually call the center of the hyperbola, the center of the ellipse. These aren't actually points on the conic section, like the vertex actually is part on the parabola, but it serves the same purpose as you allow h to get bigger and move to the right as h goes smaller and moves to the left. As k gets bigger, it goes up. As k gets smaller, it goes down. So we can set it back equal to zero. And we see our hyperbola goes back to where it was. Well, what about those parameters a and b, like you see with the ellipse, because I, again, whoops, what is that going on there? Try that again. b squared, like so if we put those into play here. At the moment a and b are set equal to one as a gets bigger, you can see that the graph is horizontally stretching out. You know, it's getting ripped away from the center of the hyperbola as a gets smaller. So the coefficient underneath the x actually affects the horizontal. It's a horizontal stretch or compression. Same thing with the b value. As b gets bigger, you're getting a vertical stretch. As b gets smaller, you're getting a vertical compression, like so. With regard to the hyperbola, there are some points of significance. That is the so-called vertices of these things. If we take the point h plus a comma k, and then we also take the point h minus a comma k, we actually do get these, and I'm going to make these ones be blue. Make it blue like a sleepy beauty, right? So as these points get stretched, these vertices are points on, these are points on the hyperbola. They live on the same line that the center has right here. This is if we of course have a vertical, a hyperbola, excuse me. If we switch it to an ellipse, we get this thing, but we put a negative sign in front of the x, then we get this horizontal, this vertical hyperbola in this situation. The vertices aren't quite right this time. So we need to take h k plus b plus b, and then we need to take h comma k minus b to get the vertices that time. And again, this will be true for our stretching and compressing. Now, why do we talk about hyperbolas now? Why not talk about them earlier? Well, it turns out as we've learned about rational functions, there actually is a hyperbola that we've seen previously, y equals one over x. This actually is an example of a hyperbola. It's just it's been rotated. This is actually the exact same hyperbola, but instead of being vertical or horizontal, this hyperbola is actually an oblique hyperbola. I'm going to hide these points here, but you can still see the vertices right, or you can see the center right there, and the vertices would be one one and negative one, negative one, right there. And so it turns out that many of the linear fractional rational functions we've dealt in the past are actually just translated and rotated hyperbolas. Okay, now one thing that's important about hyperbolas, as again, they really are just rational functions that have been rotated, they have asymptotes, like this graph right here has vertical asymptotes of y equals zero and x equals zero, right? Let's modify those. Pink seems like a good color, but let's make them dashed lines so that they look like asymptotes. This hyperbola has these asymptotes attached to it. It turns out that all hyperbolas have asymptotes as well. And so I want to show you what they look like going back to this vertical hyperbola right here. The asymptotes are actually found by the equation, you're going to take y minus k, you're going to take your slope m times x minus h, so they're going to go, they're points that go through the vertex, right? And you can kind of see where it should be. It's going to be kind of like something like this, but we can actually get, we don't have to guess what it is, that the asymptote is actually going to be b divided by a. So b was the number below the y coordinate, a is the number below the x coordinate, so it's rise over run. And so you can now see this matches up perfectly. The other one will look exact same, but it has negative slope, like so. And so every hyperbola has this set of asymptotes. And when you zoom out far away from your hyperbola, that is you can't tell the difference between the asymptotes and the hyperbola itself. It looks like just this big x on the screen here. And that's because that's what things do when they approach their asymptotes from on the grand scale. You can't see the difference. That's why I say that a hyperbola is not really two parabolas looking at each other. It just kind of looks that way on the small scale on the grand scale, though hyperbolas have an asymptotic behavior very different than a parabola. So this is the asymptotes of a vertical hyperbola. If we go back to the horizontal hyperbola, you'll see that with the same center h and k in the same parameters a and b, the same horizontal and vertical radii, the asymptotes actually turn out to be the exact same. So for any pair of asymptotes, there's actually two hyperbolas associated to it. There's the horizontal hyperbola, and then there's also this vertical hyperbola that sits in between them. And so that then talks about the three different types of conic sections, parabolas, ellipses, and hyperbolas. I do also want to mention one last thing that a hyperbola also has a notion of a focus. You have one focus over here and one focus right here, in which case there's a nice reflective property that if you have any particle that's going towards one, one of the foci, once it hits the hyperbola, it'll actually bounce off and go towards the other focus. And there are some interesting engineering applications of this focal principle. You could talk about like a hyperbolic telescope. The old Lauren Sonar system was based upon these things. The idea about triangulating the epicenter of an earthquake actually has to do with hyperbolas, which I don't necessarily want to go into all of that. I'm just happy with my students having a geometric appreciation of a hyperbola. So try some of these graphing questions for hyperbola in this week's quiz.