Let's find the missing values in the table for the function [tex]\( f(x) \)[/tex]. Given two points [tex]\((-6, 1)\)[/tex] and [tex]\(-4, 3\)[/tex], we need to determine the values of [tex]\( f(x) \)[/tex] at [tex]\( x = -5 \)[/tex].
We have two known points:
[tex]\[ (-6, 1) \][/tex]
[tex]\[ (-4, 3) \][/tex]
Assuming [tex]\( f(x) \)[/tex] is a linear function, it can be represented as:
[tex]\[ f(x) = mx + b \][/tex]
Step 1: Find the slope [tex]\( m \)[/tex]
The slope [tex]\( m \)[/tex] of the line passing through these two points is given by:
[tex]\[ m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \][/tex]
Substituting the known points into the slope formula:
[tex]\[ m = \frac{3 - 1}{-4 - (-6)} = \frac{2}{2} = 1 \][/tex]
So, the slope [tex]\( m \)[/tex] is 1.
Step 2: Find the y-intercept [tex]\( b \)[/tex]
We use one of the points [tex]\((-6, 1)\)[/tex] to find the y-intercept [tex]\( b \)[/tex]:
[tex]\[ f(x) = mx + b \][/tex]
Substitute [tex]\( x = -6 \)[/tex] and [tex]\( f(x) = 1 \)[/tex] into the linear equation:
[tex]\[ 1 = 1(-6) + b \][/tex]
[tex]\[ 1 = -6 + b \][/tex]
[tex]\[ b = 7 \][/tex]
So, the linear function [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = x + 7 \][/tex]
Step 3: Determine [tex]\( f(-5) \)[/tex] and [tex]\( f(-4) \)[/tex]
Plug [tex]\( x = -5 \)[/tex] into the linear function:
[tex]\[ f(-5) = -5 + 7 = 2 \][/tex]
Therefore, [tex]\( f(-5) = 2 \)[/tex].
Plug [tex]\( x = -4 \)[/tex] into the linear function:
[tex]\[ f(-4) = -4 + 7 = 3 \][/tex]
Therefore, [tex]\( f(-4) = 3 \)[/tex].
Putting the values into the table, we have:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
-6 & 1 \\
\hline
-5 & 2 \\
\hline
-4 & 3 \\
\hline
\end{tabular}
\][/tex]
So, the completed table is:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
-6 & 1 \\
\hline
-5 & 2 \\
\hline
-4 & 3 \\
\hline
\end{tabular}
\][/tex]
