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?

The implicit equation could be re-written in the form \(y=f(x)\) like this: \[y=\frac{(x^2-a^2)(-2a \pm \sqrt{(a^2-x^2)})}{3a^2+x^2}\]

  • How can this algebraic representation support your explanations to the previous questions?

  • Does this representation reveal anything else about the features of the graphs?

You might like to think about how you could describe the journey from the implicit equation to an equation in the form \(y=f(x)\).