Challenge Level

For a secret reason, my friend wants a curve which has 4 vertical asymptotes and 3 turning points.

Could you sketch him such a curve? Could you find an algebraic form for such a curve? Could you find many different curves with such properties?

My other friend wants a curve which also has 4 vertical asymptotes, but only 2 turning points. Can her needs be met algebraically?

As you consider this problem, many questions might emerge in your mind such as: "what makes one type of curve 'the same' as or 'different' from another?" or "can I satisfy requests for other numbers of asymptotes and turning points?". Why not make a note of these questions and ask your teacher, yourself or your friends to try to solve them?