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Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

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Cubics

Knowing two of the equations find the equations of the 12 graphs of cubic functions making this pattern.

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Patterns of Inflection

Find the relationship between the locations of points of inflection, maxima and minima of functions.

Cubic Spin

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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.