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 = \frac{a^{3}}{(x^{2} + a^{2})}

Differentiating

\frac{dy}{dx} = - \frac{2 a^{3} x}{(x^{2} + a^{2})^{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.