To determine which table represents a linear function, we need to check the consistency of the rate of change (the difference in [tex]\( y \)[/tex] values) for each given [tex]\( x \)[/tex] value in each table. A function is linear if the differences between consecutive [tex]\( y \)[/tex] values are constant.
Let's analyze each table step by step:
### Table 1
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 3 \\
\hline
2 & 7 \\
\hline
3 & 11 \\
\hline
4 & 15 \\
\hline
\end{array}
\][/tex]
Differences:
- [tex]\( 7 - 3 = 4 \)[/tex]
- [tex]\( 11 - 7 = 4 \)[/tex]
- [tex]\( 15 - 11 = 4 \)[/tex]
Since the differences are constant, this table represents a linear function.
### Table 2
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 3 \\
\hline
2 & 8 \\
\hline
3 & 15 \\
\hline
4 & 21 \\
\hline
\end{array}
\][/tex]
Differences:
- [tex]\( 8 - 3 = 5 \)[/tex]
- [tex]\( 15 - 8 = 7 \)[/tex]
- [tex]\( 21 - 15 = 6 \)[/tex]
The differences are not constant, so this table does not represent a linear function.
### Table 3
[tex]\[
\begin{array}{|c|c|}
\hline
$x$ & $y$ \\
\hline
1 & 3 \\
\hline
2 & 9 \\
\hline
3 & 3 \\
\hline
4 & 9 \\
\hline
\end{array}
\][/tex]
Differences:
- [tex]\( 9 - 3 = 6 \)[/tex]
- [tex]\( 3 - 9 = -6 \)[/tex]
- [tex]\( 9 - 3 = 6 \)[/tex]
The differences are not constant, so this table does not represent a linear function.
### Table 4
[tex]\[
\begin{array}{|c|c|}
\hline
$x$ & $y$ \\
\hline
1 & 3 \\
\hline
2 & 9 \\
\hline
\end{array}
\][/tex]
Differences:
- [tex]\( 9 - 3 = 6 \)[/tex]
While the differences are constant here, there are not enough points to conclusively determine it over a broader range. However, the consistency in the given points suggests it might be linear but it is not robustly confirmed.
By analyzing these tables, Table 1 clearly represents a linear function since its rate of change remains consistent across all given [tex]\( x \)[/tex]-values.
So, the table representing a linear function is Table 1.
