Picking hyperbola equation

So, we're asked to choose the equation that can represent the hyperbola graphed below. 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.

All right, let's think about it. This graph opens to the left and the right. Well, I guess the first thing we can realize is it's centered at (0, 0), so it's definitely just going to have the form ( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 ). We know that it opens to the left and the right; you can think of it as opening along the x direction. Thus, we know that the x term is going to be positive here, which tells us that the y term is going to be negative.

We know that the vertices here are 5 to the right of the center and 5 to the left of the center. Since the distance from the vertices to the center is 5 in the horizontal direction, we know that this right over here is going to be ( 5^2 ) or 25. So, we have 25, and this we don't quite know just yet; I'll just call this ( b^2 ). We don't know what ( b ) is.

Now, let's look at these choices here. The first option is ( \frac{x^2}{25} - \frac{y^2}{9} = 1 ). Well, that seems to match the pattern that I was able to generate really quickly just by looking at the graph, so I like this one.

This next one has the x term being negative, so that graph would open up and down, not to the left and the right. We can rule this out. The one over here has ( \frac{x^2}{9} ); that would imply that our x-intercepts are plus or minus 3 to the right and left of the center, not 5. Clearly, they aren't plus or minus 3, so we could rule this one out.

This one also has the y term being positive and the x term being negative, so once again, this would open up and down. We could rule that one out as well. Our first choice that we liked, which matched our pattern, we can feel pretty good about it now.

If you wanted to verify the 9, or if you wanted, 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 that this is the only one that even matches the general structure that we were able to deduce.