The Living Witch of Agnesi

This is a gif animation of the curve known as the 'Witch of Agnesi.' It will cycle 5 times showing how the curve is defined graphically.

The curve shown here was actually drawn analytically using the formula y = a3/(x2 + a2) where a is the amplitude, or height, of the curve at the origin (x=0). Note that to do something 'analytically' means to use a formula. To create it graphically, start by drawing a circle. Then pick a point along the x-axis (this point is labeled 'A' in the animation) and draw a line from the origin ('O'), also seen to be the bottom of the circle, to point A. Where this crosses the circle (point 'B') determines the Y value associated with the X value you first chose. Mark a point horizontally from point B over at the X value, labeled P(x,y). Do this for a variety of X-values and you will mark out the curve known as the 'Witch of Agnesi.' Notice that the curve never quite gets down to a Y value of zero. This is called asymptotic behavior. It gets ever closer to zero but never quite reaches it, until X = infinity.

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