Let's determine which of the given tables represents a linear function.
A function is linear if the differences between the [tex]\( y \)[/tex] values divided by the differences in the corresponding [tex]\( x \)[/tex] values (i.e., the slopes) are constant throughout.
1. 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]
Calculating the slopes:
[tex]\[
\frac{7-3}{2-1} = 4, \quad \frac{11-7}{3-2} = 4, \quad \frac{15-11}{4-3} = 4
\][/tex]
Since all the slopes are equal (4), Table 1 represents a linear function.
2. 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]
Calculating the slopes:
[tex]\[
\frac{8-3}{2-1} = 5, \quad \frac{15-8}{3-2} = 7, \quad \frac{21-15}{4-3} = 6
\][/tex]
Since the slopes (5, 7, 6) are not equal, Table 2 does not represent a linear function.
3. 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]
Calculating the slopes:
[tex]\[
\frac{9-3}{2-1} = 6, \quad \frac{3-9}{3-2} = -6, \quad \frac{9-3}{4-3} = 6
\][/tex]
Since the slopes (6, -6, 6) are not equal, Table 3 does not represent a linear function.
4. Table 4:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 3 \\
\hline
2 & 9 \\
\hline
\end{array}
\][/tex]
Calculating the slope:
[tex]\[
\frac{9-3}{2-1} = 6
\][/tex]
With only two points, the slope calculation is straightforward and it appears constant. Hence, Table 4 represents a linear function.
Hence, the tables that represent linear functions are Table 1 and Table 4.
