To determine which relationship has a zero slope, we need to analyze the given tables of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values.
A relationship has a zero slope if the [tex]\( y \)[/tex]-values are constant, meaning they do not change regardless of the [tex]\( x \)[/tex]-values.
Let's examine the first table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & 2 \\
\hline
-1 & 2 \\
\hline
1 & 2 \\
\hline
3 & 2 \\
\hline
\end{array}
\][/tex]
In this table, the [tex]\( y \)[/tex]-values are all 2. Since the [tex]\( y \)[/tex]-values do not change (they are constant), this relationship has a zero slope.
Now, let's examine the second table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & 3 \\
\hline
-1 & 1 \\
\hline
1 & -1 \\
\hline
3 & -3 \\
\hline
\end{array}
\][/tex]
In this table, the [tex]\( y \)[/tex]-values are 3, 1, -1, and -3. Since the [tex]\( y \)[/tex]-values are changing and are not constant, this relationship does not have a zero slope.
Therefore, based on our examination:
- The first relationship ([tex]\( x \)[/tex] and [tex]\( y \)[/tex] pairs from the first table) has a zero slope.
- The second relationship ([tex]\( x \)[/tex] and [tex]\( y \)[/tex] pairs from the second table) does not have a zero slope.
Thus, the relationship with the zero slope is the first one.
