The top of the bell shaped curve

This is the bell shaped curve from x ranging from -3 to 3. Note that this is NOT the normal curve, the famous workhorse of statistics. In math, "normal" usually means there is something about object whose measure is 1. This curve has a highest point at (0,1) but the normal curve has an area between the curve and the x-axis of 1. (In math, we would talk about this as the value of the definite integral from negative infinity to infinity.)

While this is not the normal curve, one calculus related attribute it shares with the normal curve is that the points of inflection are at x = -1 and x = 1.

Here is how the curve compares to the top of the unit circle from -1 to 1 and the parabola y = 1 - x² over the same range.

The curve that is very close to the same shape when the bottom points (-1, 0) and (1, 0) and top point (0, 1) are lined up the cosine function. The bell shaped curve is shown in blue dots and the cosine function in purple dashes. The most they disagree by is about .00647, a little more than six parts in one thousand. Still, they do disagree and are not the same function as far as mathematicians are concerned.