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Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

Sine Problem

In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.

Cocked Hat

Sketch the graphs for this implicitly defined family of functions.

Rational Request

Age 16 to 18 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?