Copyright © University of Cambridge. All rights reserved.
'Witch of Agnesi' printed from https://nrich.maths.org/
Sanjeet sent us this graph, showing the
three curves on the same graph:
For different values of $a$ the graphs have the equation: $$y =
{a^3\over (x^2+a^2)}$$ Differentiating $${dy\over dx} = -
{2a^3x\over (x^2+a^2)^2}.$$ 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.