To determine which table represents a linear function, we should verify if the relationship between \( x \) and \( y \) in each table follows a constant rate of change, i.e., the slope is constant.
Let's examine each table in detail:
### Table 1
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 5 \\
\hline
2 & 9 \\
\hline
3 & 5 \\
\hline
4 & 9 \\
\hline
\end{array}
\][/tex]
The changes in \( x \) and \( y \):
- From \( x = 1 \) to \( x = 2 \): Change in \( y \) = 9 - 5 = 4
- From \( x = 2 \) to \( x = 3 \): Change in \( y \) = 5 - 9 = -4
- From \( x = 3 \) to \( x = 4 \): Change in \( y \) = 9 - 5 = 4
The changes in \( y \) are not consistent. Therefore, Table 1 does not represent a linear function.
### Table 2
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -5 \\
\hline
2 & 10 \\
\hline
3 & -15 \\
\hline
4 & 20 \\
\hline
\end{array}
\][/tex]
The changes in \( x \) and \( y \):
- From \( x = 1 \) to \( x = 2 \): Change in \( y \) = 10 - (-5) = 15
- From \( x = 2 \) to \( x = 3 \): Change in \( y \) = -15 - 10 = -25
- From \( x = 3 \) to \( x = 4 \): Change in \( y \) = 20 - (-15) = 35
The changes in \( y \) are wildly inconsistent. Therefore, Table 2 does not represent a linear function.
### Table 3
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 5 \\
\hline
2 & 10 \\
\hline
3 & 20 \\
\hline
4 & 40 \\
\hline
\end{array}
\][/tex]
The changes in \( x \) and \( y \):
- From \( x = 1 \) to \( x = 2 \): Change in \( y \) = 10 - 5 = 5
- From \( x = 2 \) to \( x = 3 \): Change in \( y \) = 20 - 10 = 10
- From \( x = 3 \) to \( x = 4 \): Change in \( y \) = 40 - 20 = 20
Although the increments in \( y \) increase, they do so in a quadratic manner rather than linearly. Hence, Table 3 does not represent a linear function.
### Table 4
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -5 \\
\hline
2 & 0 \\
\hline
\end{array}
\][/tex]
The changes in \( x \) and \( y \):
- From \( x = 1 \) to \( x = 2 \): Change in \( y \) = 0 - (-5) = 5
There's only one interval to check, and it shows a constant rate of change. Thus, Table 4 represents a linear function.
Based on our detailed examination, the table that represents a linear function is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -5 \\
\hline
2 & 0 \\
\hline
\end{array}
\][/tex]
Therefore, the fourth table represents a linear function.
