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How Many Solutions?

Find all the solutions to the this equation.


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

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?