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How Many Solutions?

Find all the solutions to the this equation.

Quartics

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

Power Up

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Witch of Agnesi

Age 16 to 18
Challenge Level

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 .