Calculus How To

Witch of Agnesi

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The Witch of Agnesi is a special case of a cubic hyperbola that is roughly bell-shaped.

witch of agnesi graph 2

The Witch of Agnesi (red) and rotated 90° (black) with a = 5.

The algebraic equation is

xy2 = a2(a x).

Alternatively, the curve can be turned 90° so that it lies on the horizontal axis:

yx2 = a2(a – y).

The parametric equations are [1]:

  • x(t) = at
  • y(t) = a/(1 + t)2

The curve is named after the 18th century mathematician Maria Agnesi (1718 to 1799). It has a few surprising real life applications including as an approximation for the spectral line distribution of optical lines and x-rays. It also approximates the amount of power dissipated in resonant circuits [2].

Finding Derivatives

The coordinates x0 and y0 satisfy the equation
coordinates witch of agnesi

We can take the derivative (with the chain rule) to get:
derivative for witch of agnesi

Witch of Agnesi (x^2 + 4)y = 8 Implicit Differentiation Equation of the Tangent Line Problem
Watch this video on YouTube.

Construction of the Witch of Agnesi

construction of the witch of agnesi

Construction of the witch of Agnesi.

The curve can be constructed as follows [3]:

  1. Choose a circle of diameter a.
  2. Center the circle at (0, a/2)
  3. Choose a point A on the line y = a; draw a line from A to the origin. Mark a new point B where this line crosses the circle.
  4. If P is the point where the vertical line through P crosses the horizontal line through B, then the witch of Agnesi is the curve traced by P as A travels along the line y = a.

Why is it Called the Witch of Agnesi?

It turns out, it’s named the “witch” because of a mistake in translation [4]. Agnesi’s two-volume treatise on calculus included the curve, which she called versiera, which is Italian for “that-which-turns.” But when her work was translated into English by a Cambridge professor, he mistook the word for l’aversiera, which means “witch” in Italian.


[1] Fifty Famous Curves, Lots of Calculus Questions, And a Few Answers.
[2] Applications to the Witch. Retrieved March 7, 2022 from:
[3] The Witch of Agnesi. Retrieved March 6, 2022 from:
[4] Lienhard, J. No. 1741: THE WITCH OF AGNESI. Retrieved March 7, 2022 from:

Stephanie Glen. "Witch of Agnesi" From Calculus for the rest of us!

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