Sanjeet sent us this graph, showing the three curves on the same graph:

Witch of Agnesi

For different values of a the graphs have the equation:

y =

a³

(x²+a²)

Differentiating

= -

2a³x

(x²+a²)²

So the critical value is at (0,a). Near this point, for small negative x the gradient is positive and the function increasing and for small positive x the gradient is negative and the function decreasing so this is a maximum point.

The value of y is always positive so the entire graph lies above the x axis and the graph is symmetrical about x=0 because f(-x)=f(x).

For large x the value of y tends to zero so the line y=0 is an asymptote.