Chapter 7.3, Problem 68E

Skills and Applications

Fitting a Parabola One way to find the least squares regression parabola $y=a{x}^2+bx+c$for a set of points $\left(x_1,y_1\right),\left(x_2,y_2\right),\dots ,\left(x_n,y_n\right)$is by solving the system below for $a$, $b$, and $c$.

$\{\begin{array}{rll}nc+({\displaystyle \underset{i}{\overset{}{\sum }}}& =& 1\end{array}n{x}_i)b+\left({\displaystyle \underset{i=& 1}{\overset{n}{\sum }}{x}_{{}_i}^2}\right)a=& {\displaystyle \underset{i}{\overset{}{\sum }}}& =& 1n{y}_i& ({\displaystyle \underset{i}{\overset{}{\sum }}}& =& 1n{x}_i)c+({\displaystyle \underset{i}{\overset{}{\sum }}}& =& 1n{x}_{{}_i}^2)b+({\displaystyle \underset{i}{\overset{}{\sum }}}& =& 1n{x}_{{}_i}^3)a& =& {\displaystyle \underset{i}{\overset{}{\sum }}}& =& 1n{x}_i{y}_i\\ ({\displaystyle \underset{i}{\overset{}{\sum }}}& =& 1n{x}_{{}_i}^2)c+({\displaystyle \underset{i}{\overset{}{\sum }}}& =& 1n{x}_{{}_i}^3)b+({\displaystyle \underset{i}{\overset{}{\sum }}}& =& 1n{x}_{{}_i}^4)a& =& {\displaystyle \underset{i}{\overset{}{\sum }}}& =& 1n{x}_{{}_i}^2{y}_i$

In Exercises $67$and $68$, the sums have been evaluated. Solve the simplified system for $a$, $b$, and $c$to find the least squares regression parabola for the points. Use a graphing utility to confirm the result. (Note: The symbol ${\displaystyle \sum }$is used to denote a sum of the terms of a sequence. You will learn how to use this notation in Section $9.1$.)

$\{\begin{array}{rll}4c+40a& =& 19\\ 40b& =& -12\\ 40c++544a& =& 160\end{array}$