To determine which table represents a linear function, we need to check if the rate of change (or slope) between consecutive points is constant for each table.
Table 1:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -2 \\
\hline
2 & -6 \\
\hline
3 & -2 \\
\hline
4 & -6 \\
\hline
\end{array}
\][/tex]
Let's calculate the rate of change between each pair of points:
1. Between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[
\frac{-6 - (-2)}{2 - 1} = \frac{-4}{1} = -4
\][/tex]
2. Between [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
[tex]\[
\frac{-2 - (-6)}{3 - 2} = \frac{4}{1} = 4
\][/tex]
3. Between [tex]\( x = 3 \)[/tex] and [tex]\( x = 4 \)[/tex]:
[tex]\[
\frac{-6 - (-2)}{4 - 3} = \frac{-4}{1} = -4
\][/tex]
The rate of change is not constant; thus, Table 1 does not represent a linear function.
Table 2:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -2 \\
\hline
2 & -5 \\
\hline
3 & -9 \\
\hline
4 & -14 \\
\hline
\end{array}
\][/tex]
Let's calculate the rate of change between each pair of points:
1. Between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[
\frac{-5 - (-2)}{2 - 1} = \frac{-3}{1} = -3
\][/tex]
2. Between [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
[tex]\[
\frac{-9 - (-5)}{3 - 2} = \frac{-4}{1} = -4
\][/tex]
3. Between [tex]\( x = 3 \)[/tex] and [tex]\( x = 4 \)[/tex]:
[tex]\[
\frac{-14 - (-9)}{4 - 3} = \frac{-5}{1} = -5
\][/tex]
The rate of change is not constant; thus, Table 2 does not represent a linear function.
Table 3:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -2 \\
\hline
2 & -10 \\
\hline
3 & -18 \\
\hline
4 & -26 \\
\hline
\end{array}
\][/tex]
Let's calculate the rate of change between each pair of points:
1. Between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[
\frac{-10 - (-2)}{2 - 1} = \frac{-8}{1} = -8
\][/tex]
2. Between [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
[tex]\[
\frac{-18 - (-10)}{3 - 2} = \frac{-8}{1} = -8
\][/tex]
3. Between [tex]\( x = 3 \)[/tex] and [tex]\( x = 4 \)[/tex]:
[tex]\[
\frac{-26 - (-18)}{4 - 3} = \frac{-8}{1} = -8
\][/tex]
The rate of change is constant; thus, Table 3 represents a linear function.
Table 4:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -2 \\
\hline
2 & -4 \\
\hline
\end{array}
\][/tex]
Let's calculate the rate of change between the two points:
1. Between [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[
\frac{-4 - (-2)}{2 - 1} = \frac{-2}{1} = -2
\][/tex]
With only two points, the rate of change is constant by definition; thus, Table 4 represents a linear function.
In conclusion, the tables that represent a linear function are Table 3 and Table 4.
