To determine the value of [tex]\( b \)[/tex] in the quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] given the provided data points, we need to perform a regression analysis to fit the quadratic model to the data. Here are the steps to solve for the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
1. Define the quadratic model:
[tex]\[
f(x) = ax^2 + bx + c
\][/tex]
2. Collect the data points [tex]\( (x, f(x)) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-10 & 24 \\
\hline
-9 & 17 \\
\hline
-8 & 12 \\
\hline
-7 & 9 \\
\hline
-6 & 8 \\
\hline
-5 & 9 \\
\hline
-4 & 12 \\
\hline
\end{array}
\][/tex]
3. Fit the quadratic model to find the best-fit coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] using a least squares approach. This involves solving the system of equations derived from substituting each data point into the quadratic equation.
Based on the given solution, the coefficients have been calculated to be:
[tex]\[
a \approx 1.000, \quad b \approx 12.000, \quad c \approx 44.000
\][/tex]
Therefore, the value of [tex]\( b \)[/tex] is:
[tex]\[
\boxed{12}
\][/tex]
