Consider the symmetries of the graphs in this family, where the graphs cut the axes, and how they can be thought of as stretched circles. Relate these ideas to the equations you get by taking different values of the constants $a$ and $b$ in the equation

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

All these ideas come together to throw light on transformations of graphs and also on properties of ellipses.