Introduction to hyperbolas
Vertices & direction of a hyperbola (example 2)
- [Voiceover] So we're asked to choose the equation that can represent the hyperbola graphed below. And so this is the hyperbola graphed in blue and I encourage you to pause the video and figure out which of these equations are represented by the graph here. Alright, so let's think about it. This graph opens to the left and the right. Well, I guess, the first thing we could realize, it's centered at zero. So definitely, it's just going to have the form, x squared and y squared over two different things equaling one. And we know that it opens to the left and the right. You can think of it opens along the x-direction and so we know that the x-term is going to be positive here, which tells us that the y-term is going to be negative. And we know that the vertices here are five to the right of the center and five to the left of the center and so since the distance from the vertices to the center is five in the horizontal direction, we know that this right over here is going to be five squared or 25. And this we don't quite know, just yet. I just call this a squared. We don't know what a is. Now let's look at these choices here. X squared over 25 minus y squared over nine equals one. Well, that seems to match the pattern that I was able to generate really quickly, just looking at the graph, so I like this one. This has the x-term being negative. So this graph over here, this would open up up and down, not to the left and the right, so we could rule this out. This one over here. This has x squared over nine. That would imply that our x-intercepts are plus or minus three to the right and left of the center, not five. Clearly, they aren't plus or minus three so we could rule this one out. This one has the y-term being positive and the x-term being negative, so once again, this would open up and down. So we could rule that one out as well. And so our first choice that we like that matched our pattern, we could feel pretty good about it. Now if you wanted to verify the nine or if you want, you might want to try out some other points or solve some points, if it wasn't multiple choice but in this case, we are able to pick out. This is the only one that even matches the general structure that we were able to deduce.