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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?