To determine the coefficients of the quadratic function [tex]\(f(x)\)[/tex] in the form [tex]\(f(x) = ax^2 + bx + c\)[/tex], consider the given points [tex]\((-10, 24)\)[/tex], [tex]\((-9, 17)\)[/tex], and [tex]\((-8, 12)\)[/tex].
These points provide us with the following system of linear equations:
1. [tex]\(24 = a(-10)^2 + b(-10) + c \implies 24 = 100a - 10b + c\)[/tex]
2. [tex]\(17 = a(-9)^2 + b(-9) + c \implies 17 = 81a - 9b + c\)[/tex]
3. [tex]\(12 = a(-8)^2 + b(-8) + c \implies 12 = 64a - 8b + c\)[/tex]
This system of equations can be written as:
[tex]\[
\begin{cases}
100a - 10b + c = 24 & \quad \text{(Equation 1)} \\
81a - 9b + c = 17 & \quad \text{(Equation 2)} \\
64a - 8b + c = 12 & \quad \text{(Equation 3)}
\end{cases}
\][/tex]
Solving these equations, we find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
The solution of the system of equations produces the following coefficients:
- [tex]\(a = 1.0000000000000162\)[/tex]
- [tex]\(b = 12.00000000000029\)[/tex]
- [tex]\(c = 44.00000000000128\)[/tex]
Therefore, the value of [tex]\(b\)[/tex] is approximately [tex]\(12\)[/tex]. Thus, the correct answer is:
12
