To determine which table represents a linear function, we'll check the differences between the [tex]\( y \)[/tex]-values for each table. A table represents a linear function if the differences between consecutive [tex]\( y \)[/tex]-values are constant.
Let's examine each table step-by-step:
### Table A:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 \\
\hline
y & 10 & 9 & 7 & 4 \\
\hline
\end{array}
\][/tex]
Differences between [tex]\( y \)[/tex]-values:
[tex]\[
\begin{align*}
9 - 10 &= -1 \\
7 - 9 &= -2 \\
4 - 7 &= -3 \\
\end{align*}
\][/tex]
The differences are [tex]\(-1\)[/tex], [tex]\(-2\)[/tex], and [tex]\(-3\)[/tex], which are not constant. So, Table A does not represent a linear function.
### Table B:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 \\
\hline
y & 8 & 6 & 7 & 5 \\
\hline
\end{array}
\][/tex]
Differences between [tex]\( y \)[/tex]-values:
[tex]\[
\begin{align*}
6 - 8 &= -2 \\
7 - 6 &= 1 \\
5 - 7 &= -2 \\
\end{align*}
\][/tex]
The differences are [tex]\(-2\)[/tex], [tex]\(1\)[/tex], and [tex]\(-2\)[/tex], which are not constant. So, Table B does not represent a linear function.
### Table C:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 \\
\hline
y & 11 & 8 & 5 & 2 \\
\hline
\end{array}
\][/tex]
Differences between [tex]\( y \)[/tex]-values:
[tex]\[
\begin{align*}
8 - 11 &= -3 \\
5 - 8 &= -3 \\
2 - 5 &= -3 \\
\end{align*}
\][/tex]
The differences are [tex]\(-3\)[/tex] each time, which are constant. Thus, Table C represents a linear function.
### Table D:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 \\
\hline
y & 0 & 2 & 5 & 8 \\
\hline
\end{array}
\][/tex]
Differences between [tex]\( y \)[/tex]-values:
[tex]\[
\begin{align*}
2 - 0 &= 2 \\
5 - 2 &= 3 \\
8 - 5 &= 3 \\
\end{align*}
\][/tex]
The differences are [tex]\(2\)[/tex], [tex]\(3\)[/tex], and [tex]\(3\)[/tex], which are not constant. So, Table D does not represent a linear function.
From our step-by-step checks, we can conclude that the table representing a linear function is:
[tex]\[
\text{Table C}
\][/tex]
