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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

Whose Line Graph Is it Anyway?

Age 16 to 18 Challenge Level:
How might you rule out certain pairs of graphs and equations?

Where are the key points on each graph?

For each process, roughly what shape graph would you expect?

Which graphs or equations might be consistent with this?

Once you have matched a process, equation and graph how would you construct very convincing evidence that the match is correct?