Food for thought

## Problem

The image in the warm-up shows one of a family of functions represented by the implicit equation $(x^2+2ay-a^2)^2=y^2(a^2-x^2)$.

Try changing $a$ using the slider in the applet below.

What do you notice about other functions in the family?

Can you use a combination of the information provided by the graphical and algebraic representations to answer and explain some of the questions below?

• What is special about the points:
• $x=a$
• $x=-a$
• $x=0$
• Will each member of the family be represented by two curves? Why?

• What is the domain and range of each member of the family?