To determine which table represents a linear function, we need to check if the differences between the [tex]\( y \)[/tex]-values divided by the corresponding differences in the [tex]\( x \)[/tex]-values (the slopes between each pair of points) are constant.
We have four tables to check:
1. Table 1:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 3 \\
2 & 6 \\
3 & 12 \\
4 & 24 \\
\hline
\end{array}
\][/tex]
2. Table 2:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 2 \\
2 & 5 \\
3 & 9 \\
4 & 14 \\
\hline
\end{array}
\][/tex]
3. Table 3:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -3 \\
2 & -5 \\
3 & -7 \\
4 & -9 \\
\hline
\end{array}
\][/tex]
4. Table 4:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -2 \\
2 & -4 \\
3 & -2 \\
4 & 0 \\
\hline
\end{array}
\][/tex]
### Checking Each Table for Linearity
#### Table 1:
Calculate the slopes:
[tex]\[
\text{slope}_{12} = \frac{6 - 3}{2 - 1} = 3
\][/tex]
[tex]\[
\text{slope}_{23} = \frac{12 - 6}{3 - 2} = 6
\][/tex]
[tex]\[
\text{slope}_{34} = \frac{24 - 12}{4 - 3} = 12
\][/tex]
The slopes are not constant (3, 6, 12).
#### Table 2:
Calculate the slopes:
[tex]\[
\text{slope}_{12} = \frac{5 - 2}{2 - 1} = 3
\][/tex]
[tex]\[
\text{slope}_{23} = \frac{9 - 5}{3 - 2} = 4
\][/tex]
[tex]\[
\text{slope}_{34} = \frac{14 - 9}{4 - 3} = 5
\][/tex]
The slopes are not constant (3, 4, 5).
#### Table 3:
Calculate the slopes:
[tex]\[
\text{slope}_{12} = \frac{-5 - (-3)}{2 - 1} = \frac{-5 + 3}{1} = -2
\][/tex]
[tex]\[
\text{slope}_{23} = \frac{-7 - (-5)}{3 - 2} = \frac{-7 + 5}{1} = -2
\][/tex]
[tex]\[
\text{slope}_{34} = \frac{-9 - (-7)}{4 - 3} = \frac{-9 + 7}{1} = -2
\][/tex]
The slopes are constant (-2, -2, -2).
#### Table 4:
Calculate the slopes:
[tex]\[
\text{slope}_{12} = \frac{-4 - (-2)}{2 - 1} = \frac{-4 + 2}{1} = -2
\][/tex]
[tex]\[
\text{slope}_{23} = \frac{-2 - (-4)}{3 - 2} = \frac{-2 + 4}{1} = 2
\][/tex]
[tex]\[
\text{slope}_{34} = \frac{0 - (-2)}{4 - 3} = \frac{0 + 2}{1} = 2
\][/tex]
The slopes are not constant (-2, 2, 2).
### Conclusion
From the above calculations, we observe that only Table 3 has consistent slopes, indicating it represents a linear function. Therefore, the third table represents a linear function.
