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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?

Exploring Cubic Functions

Quadratic graphs are very familiar, but what patterns can you explore with cubics?

Patterns of Inflection

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

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.