Question $3$: Given the data: $x=[1,2,3,4,5],y=[2\mathrm{.}2,2\mathrm{.}8,3\mathrm{.}6,4\mathrm{.}5,5\mathrm{.}1]$ :

$($a$)$ Manually calculate the coefficients of a quadratic fit $y=a{x}^2+bx+c$ using the normal equations.

$($b$)$ Verify your coefficients using Python.

$($c$)$ Plot the data points and the quadratic best$-$fit curve using Python.