Witch of Agnesi — Definition, Formula & Examples

The Witch of Agnesi is a bell-shaped curve defined by the equation

y = \frac{a^3}{x^2 + a^2}

, where

a

is a positive constant that controls the curve's height and width. It is symmetric about the

y

-axis, has a maximum at

(0, a)

, and approaches the

x

-axis as

x \to \pm\infty

Given a circle of diameter $a$ centered at $(0, a/2)$ , the Witch of Agnesi is the locus of points constructed as follows: for each point $M$ on the circle, draw a horizontal line through $M$ and a vertical line through the intersection of line $OM$ (where $O$ is the origin) with the horizontal line $y = a$ . The locus of intersections of these two lines is the curve $y = \frac{a^3}{x^2 + a^2}$ , first studied systematically by Maria Gaetana Agnesi in 1748.

Key Formula

y = \frac{a^3}{x^2 + a^2}

Where:

$a$ = Positive constant equal to the curve's maximum height and half the diameter of the generating circle
$x$ = Horizontal coordinate
$y$ = Vertical coordinate (always positive)

How It Works

To work with the Witch of Agnesi, set the parameter

a

to control the curve's shape: the peak value is

y = a

x = 0

, and the curve has inflection points at

x = \pm \frac{a}{\sqrt{3}}

. The curve is always positive and never crosses the

x

-axis, though it approaches

y = 0

asymptotically. In calculus courses, it appears as a standard example for finding derivatives, locating extrema, identifying concavity changes, and evaluating improper integrals. The total area under the curve from

-\infty

\infty

equals

\pi a^2

Worked Example

Problem: For the Witch of Agnesi with

a = 2

, find the maximum value, the inflection points, and the total area under the curve.

Write the equation: Substitute

a = 2

into the formula.

y = \frac{8}{x^2 + 4}

Find the maximum: The maximum occurs at

x = 0

y(0) = \frac{8}{0 + 4} = 2

Find inflection points: Inflection points occur at

x = \pm \frac{a}{\sqrt{3}} = \pm \frac{2}{\sqrt{3}}

. Evaluate

y

there.

y\!\left(\pm\tfrac{2}{\sqrt{3}}\right) = \frac{8}{\frac{4}{3} + 4} = \frac{8}{\frac{16}{3}} = \frac{3}{2}

Compute total area: Use the known result for the area under the Witch of Agnesi.

A = \pi a^2 = \pi(2)^2 = 4\pi

Answer: The curve peaks at

(0, 2)

, has inflection points at

\left(\pm\frac{2}{\sqrt{3}},\, \frac{3}{2}\right)

, and encloses a total area of

4\pi

with the

x

-axis.

Why It Matters

The Witch of Agnesi is a classic exercise in single-variable calculus for practicing curve sketching — finding extrema, concavity, and asymptotic behavior all in one problem. Its shape also resembles the Cauchy (Lorentzian) probability distribution, making it relevant in physics and statistics whenever resonance or spectral line profiles arise.

Common Mistakes

Mistake: Assuming the curve crosses or touches the

x

-axis at some finite

x

-value.

Correction: The denominator

x^2 + a^2

is always positive, so

y > 0

for all

x

. The

x

-axis is a horizontal asymptote, not an intercept.