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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 .