To determine which table represents a linear function, we need to check whether the change in [tex]\( y \)[/tex] is proportional to the change in [tex]\( x \)[/tex] for each table.
### Table A
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
$x$ & 0 & 1 & 2 & 3 \\
\hline
$y$ & 0 & 0 & -1 & -2 \\
\hline
\end{tabular}
\][/tex]
- Difference in [tex]\( y \)[/tex]: [tex]\(0 - 0 = 0\)[/tex], [tex]\(-1 - 0 = -1\)[/tex], [tex]\(-2 - (-1) = -1\)[/tex]
- Change in [tex]\( x \)[/tex] is consistent (1 unit).
The ratios of change in [tex]\( y \)[/tex] to change in [tex]\( x \)[/tex] are: [tex]\( \frac{0}{1}, \frac{-1}{1}, \frac{-1}{1} \)[/tex]. These ratios are not the same for all intervals.
### Table B
[tex]\[
\begin{tabular}{|l|l|l|l|l|}
\hline
$x$ & 0 & 1 & 2 & 3 \\
\hline
$y$ & 6 & 12 & 18 & 24 \\
\hline
\end{tabular}
\][/tex]
- Difference in [tex]\( y \)[/tex]: [tex]\(12 - 6 = 6\)[/tex], [tex]\(18 - 12 = 6\)[/tex], [tex]\(24 - 18 = 6\)[/tex]
- Change in [tex]\( x \)[/tex] is consistent (1 unit).
The ratios of change in [tex]\( y \)[/tex] to change in [tex]\( x \)[/tex] are: [tex]\( \frac{6}{1}, \frac{6}{1}, \frac{6}{1} \)[/tex]. These ratios are the same for all intervals, indicating a linear function.
### Table C
[tex]\[
\begin{tabular}{|l|l|l|l|l|}
\hline
$x$ & 0 & 1 & 2 & 3 \\
\hline
$y$ & 3 & 2 & 0 & 1 \\
\hline
\end{tabular}
\][/tex]
- Difference in [tex]\( y \)[/tex]: [tex]\(2 - 3 = -1\)[/tex], [tex]\(0 - 2 = -2\)[/tex], [tex]\(1 - 0 = 1\)[/tex]
- Change in [tex]\( x \)[/tex] is consistent (1 unit).
The ratios of change in [tex]\( y \)[/tex] to change in [tex]\( x \)[/tex] are: [tex]\( \frac{-1}{1}, \frac{-2}{1}, \frac{1}{1} \)[/tex]. These ratios are not the same for all intervals.
### Table D
[tex]\[
\begin{tabular}{|l|l|l|l|l|}
\hline
$x$ & 0 & 1 & 2 & 3 \\
\hline
$y$ & 11 & 7 & 5 & 4 \\
\hline
\end{tabular}
\][/tex]
- Difference in [tex]\( y \)[/tex]: [tex]\(7 - 11 = -4\)[/tex], [tex]\(5 - 7 = -2\)[/tex], [tex]\(4 - 5 = -1\)[/tex]
- Change in [tex]\( x \)[/tex] is consistent (1 unit).
The ratios of change in [tex]\( y \)[/tex] to change in [tex]\( x \)[/tex] are: [tex]\( \frac{-4}{1}, \frac{-2}{1}, \frac{-1}{1} \)[/tex]. These ratios are not the same for all intervals.
### Conclusion
Only Table B shows a consistent ratio of the change in [tex]\( y \)[/tex] to change in [tex]\( x \)[/tex] across all intervals. Therefore, Table B represents a linear function.
