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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. ### 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. ### More Parabolic Patterns

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# Cubic Spin

##### Age 16 to 18 Challenge Level:

Prove that the graph of the polynomial

$f(x) = x^3 - 6x^2 +9x +1$

has rotational symmetry, find the centre of rotation and re-write the equation of the graph in terms of new co-ordinates $(u,v)$ with the origin of the new co-ordinate system at the centre of rotation.

Do the same for the graph of the function $g(x) = 2x^3 + 3x^2 +5x +4$ .

Prove that the graphs of all cubic polynomials have rotational symmetry.