Challenge Level

Why do this problem?

The problem gives practice in the usual techniques for cuve sketching (considering symmetry, finding turning points, looking for asymptotes). It also introduces the idea of a family of curves.

Possible approach

Suggest different members of the class sketch the different graphs (for $a=1$, $2$ and $3$). Have a class discussion about the results they find.

Key question

Will the graphs have a similar shape for all values of $a$?

What about negative values of $a$?

Possible extension

If the class can differentiate simple functions defined parametrically or implicitly then they could also try: Squareness and Folium of Descartes .