Hey, everyone, and welcome back. So, up to this point, we have talked about circles, ellipses, and parabolas. And we're now going to move on to the hyperbola shape in this series on conic sections. Now, in my opinion, the hyperbola is both the most unique and oftentimes the most difficult of all four shapes that we've covered. The reason for this is that there's a lot to remember about hyperbolas. But don't sweat it, because over the course of this video and the next few videos on conic sections, I think you're going to find that hyperbolas actually correlate very similarly to a lot of concepts that we've already learned. So, without further ado, let's get right into this. The equation for hyperbolas is nearly identical to the equation for an ellipse. The only difference is there is a minus sign rather than a plus sign.

So, if we take a look at the horizontal hyperbola, for example, notice this equation for the hyperbola, it's very similar to what we've learned previously about the ellipse. The only real difference is that we have a plus sign for the ellipse and a minus sign for the hyperbola. Even though the equations for the ellipse and hyperbola are similar, the shapes are actually quite different. Visually, a hyperbola appears as two parabolas that are facing away from each other. Now, something else that I'll mention, this ellipse that we see drawn on this graph, you're not going to actually need to ever draw this ellipse when dealing with hyperbolas. This is just here so you can see the similarities between the ellipse equation and the hyperbola equation and what the a and b values really mean. Now, looking at this horizontal hyperbola, notice that we can see what the a value is for the ellipse that we drew here. The a value is 3 units long. We've talked about in previous videos how a represents the semi-major axis of an ellipse, whereas b, which is 2 units long, represents the semi-minor axis.

Now, when it comes to the hyperbola, this a value tells you the distance to get from the center of the hyperbola to either of the two curves. As you can see, this is the shortest distance to get to this curve and the shortest distance to get to that curve. Now, this b value is a little bit less intuitive when it comes to the hyperbola because we know that this b value describes the semi-minor axis for the ellipse. But for the hyperbola, it actually serves a bit of a different purpose. It can oftentimes help in describing the height of the hyperbola, but it's also going to be very critical to use this b value when it comes to graphing the hyperbolas, which we'll talk about in later videos.

So, this is the main idea behind the horizontal hyperbola, and this is what the equation would look like if we use this a and b value because, remember, you have to square both the terms in the denominator. But now let's take a look at the vertical hyperbola. Notice for the vertical hyperbola, we have an a value of 3, except the a value is now up and down to get to either of the curves for the hyperbola rather than left and right. And notice how the b value, which is 2 units long, is now left and right as opposed to being up and down.

Now, for the ellipse, we've talked about how when you have a vertical ellipse that the semi-major axis squared is going to be underneath the y value this time, and the semi-minor axis would be under the x value. We'll notice for the hyperbola, we have a similar situation, where we have the a squared underneath the y squared and the b squared underneath the x squared. So basically, all we did was take the x and the y and switch them in this version of the equation. So the equation for the vertical hyperbola is going to be: y2 9 - x2 4 = 1

And, again, this is because you have to square the a and the b value when you put them in this equation. Now, there's one more big difference I want to mention between the ellipse and the hyperbola, and that is the major axis a. When dealing with the ellipse, we're used to this a value always being the largest value that we see. Well, when it comes to the hyperbola, the a is actually the first value that you see rather than the largest. So, for these equations in this example, we saw that the a was the biggest number, but this a is not always going to be biggest for the hyperbola. This a is simply always going to come first in the equation.

Now you may have felt like this was a lot of information, and it was. But let's actually try an example to see if we can really solidify this altogether. So, in this example, we have some equations for hyperbolas, and we're asked to match each equation with the corresponding graph. We're going to start by looking at this first equation, equation A. One of the first things I notice is that we have the y squared in front, and remember, this front term that you see is always going to be the a squared term. So you can see that a squared is equal to 16, this number here, which means that a is the square root of 16, which is 4. So, what I see here is that we have a y squared, and our a value is 4. Since we have the y squared in front, that means we're going to have a vertical hyperbola, and our a value is going to be 4. Now, if I look at these three graphs, the only vertical hyperbola I see is this one on the far right side. So if I take a look at graph number 3 here, I can see that our a value is 4. So if I start from the center, I can go 1, 2, 3, 4 units up, and 1, 2, 3, 4 units down. This does check out, which means that graph number 3 is going to match with equation A. But now let's take a look at equation B. For equation B, I can see that our a squared value, which always comes first, is going to be 4, which means a is going to be the square root of 4, which is 2. So, you can see that we have an x squared up front, which means we're going to have a horizontal hyperbola, and I can see that our a value is 2. Now, if I go ahead and start from the center of this hyperbola on graph number 2, I can go 1, 2 units to the right, and 1, 2 units to the left. Well, this is a problem, because notice that our a values go outside of the curve. So this is not the correct hyperbola that we're looking for. But now, let's try graph number 1. If we draw out our a value, we can start in the center and go 1, 2 units to the right and 1, 2 units to the left. And this lands us directly where our curve is, so this graph does check out. Now, just by process of elimination, it's clear to see that equation C is going to match with graph 2. But let's actually see if we can understand why. So I can see that the first term that we have here is 1, and this is underneath x squared. X squared means we're going to have a hyperbola that opens left and right, and this one means that our a squared value is equal to 1. Meaning a is going to be the square root of 1, which is just 1. So, if I start in the center of this hyperbola that we have here, I can go 1 unit to the right and 1 unit to the left. This is going to be our a values, and that does match with the curve that we have here. So we can see that equation C matches with graph 2. This is the main idea behind the graphing and equation of hyperbolas. Hope you found this video helpful, and thanks for watching.