To determine the value of [tex]\( b \)[/tex] in the quadratic function [tex]\( f(x) = a x^2 + b x + c \)[/tex], we need to form and solve a system of equations based on the values given in the table.
The quadratic function can be represented as:
[tex]\[ f(x) = a x^2 + b x + c \][/tex]
Using three points from the table, we can form three equations.
Given the three points:
[tex]\[
\begin{align*}
(-4, 7) \\
(-3, 6) \\
(-2, 7)
\end{align*}
\][/tex]
1. For [tex]\( f(-4) = 7 \)[/tex]:
[tex]\[ a(-4)^2 + b(-4) + c = 7 \][/tex]
[tex]\[ 16a - 4b + c = 7 \tag{1} \][/tex]
2. For [tex]\( f(-3) = 6 \)[/tex]:
[tex]\[ a(-3)^2 + b(-3) + c = 6 \][/tex]
[tex]\[ 9a - 3b + c = 6 \tag{2} \][/tex]
3. For [tex]\( f(-2) = 7 \)[/tex]:
[tex]\[ a(-2)^2 + b(-2) + c = 7 \][/tex]
[tex]\[ 4a - 2b + c = 7 \tag{3} \][/tex]
We now have a system of linear equations:
[tex]\[
\begin{align*}
16a - 4b + c &= 7 \tag{1} \\
9a - 3b + c &= 6 \tag{2} \\
4a - 2b + c &= 7 \tag{3}
\end{align*}
\][/tex]
To solve this system, let's eliminate [tex]\( c \)[/tex] by subtracting equation (2) from equation (1), and equation (3) from equation (2):
First subtract equation (2) from equation (1):
[tex]\[
(16a - 4b + c) - (9a - 3b + c) = 7 - 6
\][/tex]
[tex]\[
7a - b = 1 \tag{4}
\][/tex]
Now subtract equation (3) from equation (2):
[tex]\[
(9a - 3b + c) - (4a - 2b + c) = 6 - 7
\][/tex]
[tex]\[
5a - b = -1 \tag{5}
\][/tex]
We now have two simpler equations:
[tex]\[
\begin{align*}
7a - b &= 1 \tag{4} \\
5a - b &= -1 \tag{5}
\end{align*}
\][/tex]
Subtract equation (5) from equation (4):
[tex]\[
(7a - b) - (5a - b) = 1 - (-1)
\][/tex]
[tex]\[
2a = 2
\][/tex]
[tex]\[
a = 1
\][/tex]
Substituting [tex]\( a = 1 \)[/tex] into equation (4):
[tex]\[
7(1) - b = 1
\][/tex]
[tex]\[
7 - b = 1
\][/tex]
[tex]\[
b = 6
\][/tex]
Therefore, the value of [tex]\( b \)[/tex] is [tex]\( 6 \)[/tex].
