To determine which table represents a linear function, we need to check if the change in [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] remains constant. Let's analyze each table in detail:
1. 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]
- Change from [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( \frac{9 - 5}{2 - 1} = 4 \)[/tex]
- Change from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( \frac{5 - 9}{3 - 2} = -4 \)[/tex]
- Change from [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]: [tex]\( \frac{9 - 5}{4 - 3} = 4 \)[/tex]
The changes are not consistent. Hence, Table 1 does not represent a linear function.
2. 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]
- Change from [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( \frac{10 - (-5)}{2 - 1} = 15 \)[/tex]
- Change from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( \frac{-15 - 10}{3 - 2} = -25 \)[/tex]
- Change from [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]: [tex]\( \frac{20 - (-15)}{4 - 3} = 35 \)[/tex]
The changes are not consistent. Hence, Table 2 does not represent a linear function.
3. 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]
- Change from [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( \frac{10 - 5}{2 - 1} = 5 \)[/tex]
- Change from [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]: [tex]\( \frac{20 - 10}{3 - 2} = 10 \)[/tex]
- Change from [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]: [tex]\( \frac{40 - 20}{4 - 3} = 20 \)[/tex]
The changes are not consistent. Hence, Table 3 does not represent a linear function.
4. Table 4:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -5 \\
\hline
2 & 0 \\
\hline
\end{array}
\][/tex]
- Change from [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\( \frac{0 - (-5)}{2 - 1} = 5 \)[/tex]
There is only one interval to check, and since there are no contradictions, the change is consistent. Hence, Table 4 represents a linear function.
Conclusion: The table that represents a linear function is Table 4.
