Hyperbolas

Earlier this week, we learned about hyperbolas. Hyperbolas look like two parabolas facing opposite directions. Here are the equations for the two types of hyperbolas:

The best way to explain hyperbolas would be with an example, so here is the equation of an example we will walk through:

Already, we know that the equation above represents a vertical hyperbola because the y term precedes the x term.

The center of a hyperbola is given by (h, k), but in the equation above, there is neither an h nor a k, so we will set them to 0, making the center (0, 0).

The vertices of a hyperbola are found by adding and subtracting a from the dominant term in the ordered pair of the center. In this case, the dominant term is the y term, which is 0. This gives us the vertices (0, 5) and (0, -5) because the square root of 25 is 5.

The foci are found by adding and subtracting c from the dominant term in the ordered pair of the center, which is the y term again. The value of c squared is the sum of a squared and b squared. After doing this calculation, we find that c = 13 for this particular equation. This gives us the foci (0, 13) and (0, -13).

Finally, we must find the asymptotes. We will use the formulas m = b/a and m = -b/a, where m is the slope. Doing this gives us the asymptotes y = ±(5/12)x.

Now, we have finished finding the information for this hyperbola and can graph it!