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

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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.

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Cocked Hat

Sketch the graphs for this implicitly defined family of functions.

Witch of Agnesi

Stage: 5 Challenge Level: Challenge Level:1

Why do this problem?
The problem gives practice in the usual techniques for cuve sketching (considering symmetry, finding turning points, looking for asymptotes). It also introduces the idea of a family of curves.

Possible approach
Suggest different members of the class sketch the different graphs (for $a=1$, $2$ and $3$). Have a class discussion about the results they find.

Key question
Will the graphs have a similar shape for all values of $a$?

What about negative values of $a$?

Possible extension
If the class can differentiate simple functions defined parametrically or implicitly then they could also try: Squareness and Folium of Descartes .