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Cubic Spin

Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational symmetry. Do graphs of all cubics have rotational symmetry?

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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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Parabolic Patterns

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.


Stage: 4 and 5 Challenge Level: Challenge Level:1
The equation $${x^2\over a^2} + {y^2\over b^2}= 1$$ gives a family of ellipses if you take different values of the constants $a$ and $b$.