To determine which table represents a linear function with a slope of zero, we need to consider what it means for a function's slope to be zero. A slope of zero indicates that the line is horizontal, meaning that [tex]\( y \)[/tex] does not change as [tex]\( x \)[/tex] changes. Therefore, all [tex]\( y \)[/tex]-values must be the same across all [tex]\( x \)[/tex]-values.
Let’s analyze each table:
### First Table
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-5 & 5 \\
\hline
-4 & 5 \\
\hline
-3 & 5 \\
\hline
-2 & 5 \\
\hline
-1 & 5 \\
\hline
\end{array}
\][/tex]
In the first table, all the [tex]\( y \)[/tex]-values are 5, regardless of the [tex]\( x \)[/tex]-values. Therefore, the [tex]\( y \)[/tex]-values are constant, which indicates a slope of zero.
### Second Table
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -5 \\
\hline
2 & -4 \\
\hline
3 & -3 \\
\hline
4 & -2 \\
\hline
5 & -1 \\
\hline
\end{array}
\][/tex]
In the second table, the [tex]\( y \)[/tex]-values change as the [tex]\( x \)[/tex]-values change. This means the slope is not zero since [tex]\( y \)[/tex] is not constant.
### Third Table
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-5 & 5 \\
\hline
-4 & 4 \\
\hline
-3 & 3 \\
\hline
-2 & 2 \\
\hline
\end{array}
\][/tex]
In the third table, the [tex]\( y \)[/tex]-values also change with the [tex]\( x \)[/tex]-values. Therefore, the slope is not zero because [tex]\( y \)[/tex] is not constant.
### Conclusion
The only table where [tex]\( y \)[/tex] is constant (and the slope is zero) is the first table. Hence, the data in the first table represents a linear function that has a slope of zero.
