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Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

Witch of Agnesi

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

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Patterns of Inflection

Age 16 to 18
Challenge Level

A point of inflection of a curve $y=f(x)$ is a point at which the second derivative $\frac{d^2y}{dx^2}$ changes sign.

Geometrically, you can think of a point of inflection as a point where the tangent to the curve crosses the curve.
Points of inflection need not also be stationary points (first derivative also zero), although they might be.