To determine which of the given tables represents a linear function, we need to examine the changes in \( y \) for each increment in \( x \). A linear function has a constant rate of change (slope), and this means the difference between consecutive \( y \) values should be the same.
We have four different tables of \( x \) and \( y \) values.
Let's analyze each table:
1. First Table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 3 \\
\hline
2 & 6 \\
\hline
3 & 12 \\
\hline
4 & 24 \\
\hline
\end{array}
\][/tex]
Differences in \( y \) values:
[tex]\[
6 - 3 = 3 \\
12 - 6 = 6 \\
24 - 12 = 12
\][/tex]
Since the differences are not constant, this does not represent a linear function.
2. Second Table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 2 \\
\hline
2 & 5 \\
\hline
3 & 9 \\
\hline
4 & 14 \\
\hline
\end{array}
\][/tex]
Differences in \( y \) values:
[tex]\[
5 - 2 = 3 \\
9 - 5 = 4 \\
14 - 9 = 5
\][/tex]
Since the differences are not constant, this does not represent a linear function.
3. Third Table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -3 \\
\hline
2 & -5 \\
\hline
3 & -7 \\
\hline
4 & -9 \\
\hline
\end{array}
\][/tex]
Differences in \( y \) values:
[tex]\[
-5 - (-3) = -2 \\
-7 - (-5) = -2 \\
-9 - (-7) = -2
\][/tex]
Since the differences are constant (\(-2\)), this represents a linear function.
4. Fourth Table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -2 \\
\hline
2 & -4 \\
\hline
3 & -2 \\
\hline
4 & 0 \\
\hline
\end{array}
\][/tex]
Differences in \( y \) values:
[tex]\[
-4 - (-2) = -2 \\
-2 - (-4) = 2 \\
0 - (-2) = 2
\][/tex]
Since the differences are not constant, this does not represent a linear function.
Based on the above analysis, the only table that represents a linear function is the third table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -3 \\
\hline
2 & -5 \\
\hline
3 & -7 \\
\hline
4 & -9 \\
\hline
\end{array}
\][/tex]
