You are asked to sketch the graphs in this family not by plotting and joining points but by considering the general features of the graph of the rectangular hyperbola

xy = 1

and of transformations of this graph.

When you make your own pattern of hyperbolae using the equation

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

you might like to try making the same transformations as those used in the example Ellipses where the family of equations is given by

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 .$

See the challenge Ellipses here.