Witch of Agnesi

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

 

Image

For different values of $a$ the graphs have the equation: $$y=\frac{a^3}{(x^2+a^2)}$$ Differentiating $$\frac{dy}{dx}=-\frac{2a^{3}x}{(x^2+a^2{)}^2}.$$ 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.