Approximation - how to graph a probability density function?

Question

I was wondering how to graph a probability density function with the following information: https://prnt.sc/lpvnxm.

Answer 1

$$ f(x) = \frac{1}{ \sqrt{ 2 \pi \sigma }} \exp \bigg( - \frac{ (x - \mu )^2 }{2 \sigma^2 } \bigg)$$

Since $e^x \approx 1+x $

the Quadratic approximation must be

$$ Q(x) = K \bigg(1 - \frac{ (x - \mu )^2 }{2 \sigma^2} \bigg)$$ Only valid for $|x - \mu | < \sqrt2 \sigma $

$K$ is a normalization constant that you can calculate by making the requirement

$$ \int _{\mu- \sqrt2 \sigma} ^{\mu-+\sqrt2 \sigma} Q(x)dx = 1$$