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Witch of Agnesi

The Witch of Agnesi is a smooth bell-shaped plane curve in Calculus II, often written with parametric equations and also as a Cartesian equation. It is symmetric about the y-axis and shows up in parametric curve work.

Last updated July 2026

What is the witch of Agnesi?

The Witch of Agnesi is a specific plane curve you may meet in Calculus II when studying parametric equations. It is a smooth, bell-shaped curve that can be described by

x = 2a tan(t) y = 2a cos^2(t) / (1 + tan^2(t))

with the parameter t running over all real numbers. The constant a controls the size of the curve, so changing a stretches or shrinks the graph.

If you convert it to rectangular form, the curve becomes

y = 8a^3 / (4a^2 + x^2).

That form makes the shape easier to recognize. Since x appears squared in the denominator, the graph is symmetric about the y-axis, and the y-values get smaller as |x| gets larger.

The curve has a peak at x = 0, where y = 2a. From there it falls off on both sides, which is why it looks like a rounded hump or bell. It does not have a sharp corner, cusp, or loop. A common mistake is to assume the word "witch" means something spooky or mystical, but the name comes from a mistranslation of the Italian word versiera, which referred to a curve.

In Calc II, the main skill is not memorizing the name, but recognizing how the same curve can appear in two forms. The parametric form comes from the idea of tracing points with a parameter, while the rectangular form comes from eliminating the parameter and rewriting the relation between x and y. That translation between forms is the real math move behind this term.

Why the witch of Agnesi matters in Calculus II

The Witch of Agnesi matters because it gives you a concrete example of how parametric equations describe curves that are not always easiest to write as y = f(x) right away. In Calculus II, that kind of example shows up when you are learning how to sketch a parametric curve, convert it to rectangular form, or identify symmetry from an equation.

It also reinforces a few core Calc II habits. First, you look at the structure of the equation and notice how x and y depend on a parameter. Then you ask whether the curve has a shape you can predict before graphing it. Here, the squared x in the denominator tells you the curve is even, so the graph mirrors across the y-axis.

This curve is also useful as a bridge between algebra and calculus thinking. You can study its shape, compare it with other familiar curves, and use the parameter a to see how a formula changes the graph. In some classes, it also comes up as a classic named curve, which makes it a good reference point when your instructor talks about special parametric curves or symmetry.

Keep studying Calculus II Unit 7

Official unit cheatsheet

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Unit 7 study guide

How the witch of Agnesi connects across the course

Parametric Equations

The Witch of Agnesi is usually introduced through parametric equations, where both x and y depend on a parameter t. That makes it a good example of how a curve can be traced point by point instead of being written directly as y = f(x). In Calc II, you may be asked to graph the curve from its parameterization or eliminate the parameter.

Rectangular Form

One common task with the Witch of Agnesi is converting from parametric form to rectangular form. The rectangular equation y = 8a^3 / (4a^2 + x^2) shows the curve more plainly and makes symmetry easier to spot. This is the same skill you use when rewriting other parametric curves in terms of x and y only.

Symmetry

The Witch of Agnesi is symmetric about the y-axis because x appears squared in the rectangular equation. That means replacing x with -x gives the same y-value. In Calculus II, symmetry checks save time when graphing curves and can help you predict the overall shape before making a table of points.

cusps

The Witch of Agnesi does not have cusps, which makes it a useful contrast term. Cusps are sharp points where a curve changes direction abruptly, while the Witch of Agnesi is smooth everywhere on its graph. If you are classifying parametric curves, that difference matters a lot.

Is the witch of Agnesi on the Calculus II exam?

A quiz or problem-set question might give you the parametric equations and ask you to identify the curve, convert it to rectangular form, or sketch its shape. You may also need to use the equation to find symmetry, the highest point, or the effect of changing the parameter a. If your instructor mixes curve names into a matching question, the safest move is to connect the formula to the bell-shaped graph and the y-axis symmetry. On homework, this term usually appears in a section on parametric equations rather than as a standalone memorization item.

Key things to remember about the witch of Agnesi

  • The Witch of Agnesi is a smooth, bell-shaped curve that appears in Calculus II, especially in the unit on parametric equations.

  • Its parametric form uses a parameter t, but it can also be rewritten in rectangular form as y = 8a^3 / (4a^2 + x^2).

  • The curve is symmetric about the y-axis because x is squared in the rectangular equation.

  • The constant a changes the scale of the curve, and the peak occurs at x = 0.

  • Do not be fooled by the name, because "witch" comes from a mistranslation of versiera, not from any magical meaning.

Frequently asked questions about the witch of Agnesi

What is the Witch of Agnesi in Calculus II?

It is a named plane curve that you study as an example of a parametric curve. In rectangular form, it is y = 8a^3 / (4a^2 + x^2), which gives it a smooth hump shape. The curve is symmetric about the y-axis and is often used when discussing curve sketching and converting between forms.

How do you graph the Witch of Agnesi?

Start with the rectangular equation and notice that x is squared, so the graph is symmetric about the y-axis. The highest point is at x = 0, and then the curve falls off on both sides. If your class gives the parametric form, you may first eliminate the parameter and then sketch the resulting bell-shaped curve.

Why is it called the Witch of Agnesi?

The name comes from a mistranslation of the Italian word versiera, which referred to a curve. It has nothing to do with witches in the spooky sense. The curve is named after Maria Gaetana Agnesi, the mathematician associated with it.

Is the Witch of Agnesi a cusped or looping curve?

No, it is smooth and does not have cusps or loops. That makes it different from many other parametric curves you may study in Calculus II. Its shape is a single rounded hump, which is one reason it is easy to recognize once you know the formula.