To determine which table represents a linear function, we can check if the differences between consecutive [tex]\( y \)[/tex]-values are consistent. A table represents a linear function if [tex]\( y \)[/tex] changes by a constant amount as [tex]\( x \)[/tex] increases by 1.
Let's go through each table:
Table 1:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 1 \\
\hline
1 & 3 \\
\hline
2 & -3 \\
\hline
3 & -1 \\
\hline
\end{array}
\][/tex]
Calculate the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[
3 - 1 = 2
\][/tex]
[tex]\[
-3 - 3 = -6
\][/tex]
[tex]\[
-1 - (-3) = 2
\][/tex]
The differences are [tex]\( 2, -6, 2 \)[/tex], which are not consistent. Therefore, Table 1 does not represent a linear function.
Table 2:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 0 \\
\hline
1 & 1 \\
\hline
2 & 4 \\
\hline
3 & 9 \\
\hline
\end{array}
\][/tex]
Calculate the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[
1 - 0 = 1
\][/tex]
[tex]\[
4 - 1 = 3
\][/tex]
[tex]\[
9 - 4 = 5
\][/tex]
The differences are [tex]\( 1, 3, 5 \)[/tex], which are not consistent. Therefore, Table 2 does not represent a linear function.
Table 3:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 3 \\
\hline
1 & 1 \\
\hline
2 & -1 \\
\hline
3 & -3 \\
\hline
\end{array}
\][/tex]
Calculate the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[
1 - 3 = -2
\][/tex]
[tex]\[
-1 - 1 = -2
\][/tex]
[tex]\[
-3 - (-1) = -2
\][/tex]
The differences are [tex]\( -2, -2, -2 \)[/tex], which are consistent. Therefore, Table 3 represents a linear function.
Table 4:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 0 \\
\hline
1 & -1 \\
\hline
2 & 4 \\
\hline
3 & -9 \\
\hline
\end{array}
\][/tex]
Calculate the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[
-1 - 0 = -1
\][/tex]
[tex]\[
4 - (-1) = 5
\][/tex]
[tex]\[
-9 - 4 = -13
\][/tex]
The differences are [tex]\( -1, 5, -13 \)[/tex], which are not consistent. Therefore, Table 4 does not represent a linear function.
Based on our analysis, the table that represents a linear function is:
[tex]\[
\boxed{3}
\][/tex]
