Take ex...
Square the x, and the function is now positive on the left side.
You get an extremely narrow parabola variant. Here's a parabola in green for comparison. (The next two images are just for illustration, not part of the process.)
(Technically, it's the exponential of a parabola. It's ex2 instead of x2. If you ask me, that counts as a parabola variant. But it grows much faster.)
Now negate the exponent.
Voila.
That's the rule. It's an upside-down parabola for an exponent.
The rest of the famed formula is tweaking—specifying the unit size via the mean (\(\mu\)) and the standard deviation (\(\sigma\)). It looks complicated, but this is a lot like describing a parabola with \( (y-k) = 4p(x-h)^2 \) instead of just \(x^2\). The second gives you the foundational idea, while the first incorporates adjustments.
\[\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}\]
Great to have on hand as a reference, but we already have the essential bell curve just from two modifications to basic exponential growth. We square the exponent and then negate the exponent.
\[e^x \rightarrow e^{-x^2} \] Oh, and if the e is confusing, we could have started with any example of exponential growth. For example, we could use a base of 2. (A base multiplier of 2 sets a slightly slower growth rate than with e.) The picture would look much the same.\[2^x \rightarrow 2^{-x^2} \]
This time I'll leave out the two graphs comparing with a basic parabola, because they weren't really part of the process anyway, and they look the same. And remember, there's nothing very special about 2 or e here. Any real number can be turned into a bell curve by exponentiating, squaring the exponent, and negating the exponent.\[2 \rightarrow 2^x \rightarrow 2^{x^2} \rightarrow 2^{-x^2} \]
\[3 \rightarrow 3^x \rightarrow 3^{x^2} \rightarrow 3^{-x^2} \]
\[5.8316 \rightarrow 5.8316^x \rightarrow 5.8316^{x^2} \rightarrow 5.8316^{-x^2} \]
\[\pi \rightarrow \pi^x \rightarrow \pi^{x^2} \rightarrow \pi^{-x^2} \]
Below is with a base of 2.