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 the other functions in the family?

A graph of the implicit curve $(x^2+2ay-a^2)^2=y^2(a^2-x^2)$ for $a=3$

Are there any questions that you would like to ask about the images and the equation?

  • Can you sketch the curve for a value of \(a\) not visible in the applet, for example, \(a=7\)?

    • What makes this easy?
    • What makes this difficult?

  • What happens if \(a=0\)?

  • What happens if \(a<0\)?

  • Does the equation always produce two curves? Why?

  • Are the ‘end-points’ of the curves always located at \(x=-a\) and \(x=a\)? Why?

  • Is the maximum of the ‘biggest’ curve located at \((0,a)\)?

    • Where is the maximum of the ‘smaller’ curve?

  • Is the graph always symmetrical about the \(y\)-axis?

  • Can the equation be re-written in the form \(y=f(x)\)? Why might this be useful?