### Folium of Descartes

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.

### Quick Route

What is the quickest route across a ploughed field when your speed around the edge is greater?

# Patterns of Inflection

##### Stage: 5 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.