Chapter 7.5, Problem 22E

Repeat Problem 21 for the points (−1, −2), (0, 1), (1, 2), and (2, 0).

21. To find the coefficients of the parabola

     $y=a{x}^2+bx+c$

that is the “best” fit to the points (1, 2), (2, 1), (3, 1), and (4, 3), minimize the sum of the squares of the residuals

     $\begin{array}{ccc}F\left(a,b,c\right)& =& {\left(a+b+c-2\right)}^2\\ & & +{\left(4a+2b+c-1\right)}^2\\ & & +{\left(9a+3b+c-1\right)}^2\\ & & +{\left(16a+4b+c-3\right)}^2\end{array}$

by solving the system of normal equations

   $\begin{array}{ccc}{F}_a\left(a,b,c\right)=0& F_b\left(a,b,c\right)=0& F_c\end{array}\left(a,b,c\right)=0$

for a_, b_, and c. Graph the points and the parabola.