Hyperbolas can be sketched quite accurately by first locating the vertices, the foci, and the asymptotes. Starting with the axes, locate the vertices and foci. Draw a circle with its center at C, passing through the two foci. Draw lines through the vertices perpendicular to the transverse axis. This determines four points, which are corners of a rectangle. These diagonals are the asymptotes.
Using the vertices and asymptotes as guides, sketch in the hyperbola as shown in Figure 5. The hyperbola approaches the asymptotes, but never quite reaches them. Its curvature, therefore, approaches, but never quite reaches, that of a straight line.
If the lengths of the transverse and conjugate axes are known, the rectangle in Figure 5 can be drawn without
using the foci, since the rectangle's length and width are equal to these axes.
One can also draw hyperbolas by plotting points on a coordinate plane. In doing this, it helps to draw the asymptotes, whose equations are given above.