Hyperbolas

 Hyperbola is defined as a set of all points (x,y) the difference of whose distances from two distinct fixed points (foci) is constant. 

The equation is given by so (standard form)

Going horizontal: (x - h)^2  - (y - k)^2

                               a^2.           b^2 

Going vertical:     (y - k)^2  -. (x - h)^2

                               a^2.            b^2 

Foci: c^2 = a^2 + b^2

Notice how, unlike ellipses, a doesn't denote the major axis. This is due to the fact that such things are not used here, a and b stay in the same place regardless of anything. The direction in which a hyperbola goes (horizontal or vertical) depends on whether x or y is the first term. If y is the first term, the hyperbolas go along the y axis. Opposite is true if x is the leading term, 

Center (h,k) 

Vertices:

 if hyperbola is going horizontal (x leading term): (h+a,k) (h-a,k)

If hyperbola is going vertical (y is the leading term): (h, k+a)(h, k-a) 

Remember for the box that we add b to the h or k to find the boundaries.

As for the foci

If hyperbola going horizontal: (h + c, k) (h - c, k) 

If hyperbola going vertical: (h, k + c) (h, k - c)

Asymptotes

Horizontal: y=k +/- b/a (x - h)

Vertical: y = k +/- a/b (x - h)

Here's an example of his to make a hyperbola.