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?

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?