$S$

$\=\sum _{k\=1}^5{\left(f\left(x_k\right)-y_k\right)}^2$

$\=979a^2\+450ab\+110ac\+55b^2\+30bc\+5c^2-3200 a-748 b-188 c\+2642$

Define lists of abscissas and ordinates

$X\=\left[1\,2\,3\,4\,5\right]$$\stackrel{\text{assign}}{\to }$

$Y\=\left[3\,8\,15\,30\,38\right]$$\stackrel{\text{assign}}{\to }$

$f\left(x\right)\=a x^2\+b x\+c$$\stackrel{\text{assign as function}}{\to }$$f$

Define $S$, the sum of squares of deviations

$S\=\sum _{k\=1}^5{\left(f\left(X_k\right)-Y_k\right)}^2$$\stackrel{\text{assign}}{\to }$

Minimize $S$ 

$\frac{\partial }{\partial a} S\=0\,\frac{\partial }{\partial b} S\=0\,\frac{\partial }{\partial c} S\=0$

$1958a+450b+110c-3200=0,450a+110b+30c-748=0,110a+30b+10c-188=0$

$\stackrel{\text{solve}}{\to }$

$\left\{a=1\,b=\frac{16}{5}\,c=-\frac{9}{5}\right\}$

$\stackrel{\text{assign to a name}}{\to }$

$Q$

Obtain the least-squares quadratic

$\genfrac{}{}{0ex}{}{f\left(x\right)}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{Q}$ = $-\frac{9}{5}+\frac{16}{5}x+x^2$$$

Table 4.8.12(a)   Construction of least-squares line from first principles.

Launch the Curve-Fitting Assistant from the Context Panel

$X\,Y$ = $\left[1\,2\,3\,4\,5\right],\left[3\,8\,15\,30\,38\right]$ 

Table 4.8.12(b)   Least-squares quadratic via the Curve-Fitting Assistant

Tools≻Tasks≻Browse: Linear Algebra≻Visualizations≻Least Squares Plot

Table 4.8.12(c)   The Least-Squares Plot task template

Enter the data

$X\,Y≔\left[1\,2\,3\,4\,5\right]\,\left[3\,8\,15\,30\,38\right]\:$$$

Define the quadratic fitting function $f\left(x\right)$ 

$f≔x\to a x^2\+b x\+c\:$

Obtain $S$, the sum of squares of the deviations

$S≔\mathrm{sum}\left({\left(f\left(X_k\right)-Y_k\right)}^2\,k\=1..5\right)\:$

Minimize $S$ 

Use the diff command to form equations and the solve command to solve them for $a\,b$, and $c$.

$Q≔\mathrm{solve}\left(\left\{\mathrm{diff}\left(S\,a\right)\=0\,\mathrm{diff}\left(S\,b\right)\=0\,\mathrm{diff}\left(S\,c\right)\=0\right\}\,\left\{a\,b\,c\right\}\right)\:$

Insert the optimal values of $a\,b$, and $c$ into $f\left(x\right)$  

$\mathrm{eval}\left(f\left(x\right)\,Q\right)$

$-\frac{9}{5}+\frac{16}{5}x+x^2$

$\mathrm{CurveFitting}:-\mathrm{LeastSquares}\left(X\,Y\,x\,\mathrm{curve}\=f\left(x\right)\right)$

$-\frac{9}{5}+\frac{16}{5}x+x^2$

$\mathrm{Student}:-\mathrm{LinearAlgebra}:-\mathrm{LeastSquaresPlot}\left(X\,Y\,x\,\mathrm{curve}\=f\left(x\right)\,\mathrm{boxoptions}\=\left[\mathrm{color}\=\mathrm{magenta}\right]\,\mathrm{pointoptions}\=\left[\mathrm{symbol}\=\mathrm{solidcircle}\,\mathrm{symbolsize}\=15\right]\right)$

Table 4.8.12(d)   Least-squares line via the LeastSquaresPlot command