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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.


Age 14 to 18 Challenge Level:

The illustration below shows the graphs of fourteen functions.
Two of them have equations $$y=(x+6)^3-2$$ $$y=-(x-9)^3+3$$

Can you find the equations of the other twelve graphs in this pattern?

You may wish to use a graphical calculator or software such as Desmos to recreate the pattern for yourself.