To determine the type of function represented by the given table, we'll analyze the relationship between the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values. Here is the table for reference:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
y & -7 & -2 & 3 & 8 & 13 & 18 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution:
1. List the given [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & -7 \\
1 & -2 \\
2 & 3 \\
3 & 8 \\
4 & 13 \\
5 & 18 \\
\hline
\end{array}
\][/tex]
2. Calculate the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & y & \Delta y \\
\hline
0 & -7 & - \\
1 & -2 & -2 - (-7) = 5 \\
2 & 3 & 3 - (-2) = 5 \\
3 & 8 & 8 - 3 = 5 \\
4 & 13 & 13 - 8 = 5 \\
5 & 18 & 18 - 13 = 5 \\
\hline
\end{array}
\][/tex]
So, the differences between consecutive [tex]\( y \)[/tex]-values are all 5.
3. Analyze the differences:
Since the differences between consecutive [tex]\( y \)[/tex]-values are constant (5), we have a consistent rate of change.
### Conclusion:
A function that has a constant rate of change in [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] is a linear function.
Thus, the type of function represented by the table is linear.
