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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
Consider the symmetries of the graphs in this family, where the graphs cut the axes, and how they can be thought of as stretched circles. Relate these ideas to the equations you get by taking different values of the constants $a$ and $b$ in the equation $${x^2\over a^2} + {y^2\over b^2}=1.$$ All these ideas come together to throw light on transformations of graphs and also on properties of ellipses.