How do I get the function of a bell curve from the y-value of the cusp and the standard deviation? I'm trying to figure out the formula for a bell curve knowing only the y-value of the cusp and the standard deviation.
Is it supposed to be the mean?  Because I tried putting it into a calculator and the y-value of the cusp increased when I increased the standard deviation.
 A: By the 'cusp' I suppose you mean the mode, which is also the same as
the mean and the median for a normal distribution. In general, these
are values on the x-axis (which coincide for a normal distrobution).
As in the Comment, the 'y-value' must mean the value of the pdf $y = f(\mu)$
at the mean $\mu$. That is $\frac{1}{\sqrt{2\pi}\sigma}.$ So, given that  y-value, you can
solve for the standard deviation $\sigma.$ Notice that the y-value at the mode
should decrease as the standard deviation increases.
The general formula for a normal density curve (PDF) is
$$f(x|\mu,\sigma) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right].$$
When $\mu = x$ the argument of the exponential factor is $0$ so the 
exponential factor is $1$. If you know $x = \mu$ and the corresponding $y$,
then you can write the density function with no unknown constants.
Here is the graph of a normal distribution with $\mu = 10$ (mean, median, and mode), and standard deviation $\sigma = 2.$

The vertical purple line is at $x = \mu = 10$ and the horizontal
purple line is at $y = f(\mu) = \frac{1}{2\sqrt{2\pi}} =  0.1994711.$
