Hi Alyssa -
A Hyperbola can open up either left and right or it can open up and down.
Standard Form (opens left and right) = [(x-h)^2] / a^2 - ([(y-k)^2] / b^2 = 1
Standard Form (opens up and down): [(y-k)^2] / a^2 - ([(x-h)^2] / b^2 = 1
The center of the hyperbola = (h,k). The distance from (h,k) to a vertex equals a and half the length of the conjugate axis is equal to b.
So, in your problem, the center of the hyperbola equals the midpoint between the two vertices or (0,0), so now we know h=0 and k=0. We also know the hyperbola opens left and right because the line connecting the vertices is parallel to the x-axis, so we will use the Standard Form (opens left and right) above.
Finally we need to find "a" and "b". "a" equals the length of the line from the center of the hyperbola to a vertex which equals 4. "b" equals half the conjugate length or 8/2 = 4. So, we have a=4 and b = 4.
Plug these values (h,k,a and b) into the first equation and you will have your answer.