Use the formula s^2/a^2 - y^2/b^2 = 1 to graph a hyperbola. The constants "a" and "b" are useful in sketching the hyperbola -- as is another constant "c," which can be computed with the formula c^2 = a^2 + b^2. This means that for the hyperbola x^2/9 - y^2/16 = 1, you would have a = 3, b = 4 and c = 5. The distance from the origin to the vertex of each branch of the hyperbola is "a," and the distance from the origin to the focus of a branch is "c."
Draw a box, centered at the origin, between the two branches of the hyperbole. The box is "a" by "b" and just fits snugly between the branches of the hyperbola. If you extend the two diagonals of the box infinitely in both directions, you have the asymptotes of the branches of the hyperbola. As the branches of the hyperbola go out, they get closer and closer to the asymptotes but never touch them.
Mark the "a" and "c" spots and draw the box as the first steps to drawing the hyperbola. Draw the asymptotes. Draw each branch with the vertex at "a" and making the curve so each point is placed so that the difference in the distance to the foci are constant and the branch starts approaching the asymptotes as the branch goes out. When the hyperbola is drawn, erase the box and the asymptotes.