If you plot these points on a coordinate plane, you see that both vertices and foci lie on the y axis. This means that you have a vertical hyperbola, and the equation looks like this: [tex] \frac{(y-k) ^{2} }{ a^{2} } - \frac{(x-h) ^{2} }{ b^{2} } =1[/tex] where h and k are the center. When you look at your graph, the origin is dead center between the vertices. (0, 0) is our h and k. Now we need a, b, and c. a is the distance between the center and the vertices, so our a = 4, and c is the distance between the center and the foci, so our c = 5. Use these in Pythagorean's Theorem to solve for b: [tex](4) ^{2}+ b^{2}=(5) ^{2} [/tex] and [tex]16+ b^{2} =25[/tex] and b = 3. So we have all we need to do is replace all the variables. Our equation then would be this one: [tex] \frac{(y-0) ^{2} }{16} - \frac{(x-0) ^{2} }{9} =1[/tex] or, simplified, [tex] \frac{ y^{2} }{16} - \frac{ x^{2} }{9} =1[/tex]
