Find an equation in standard form for the hyperbola with vertices at (0, ±4) and foci at (0, ±5).

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: 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: and and b = 3.  So we have all we need to do is replace all the variables.  Our equation then would be this one: or, simplified,