To determine whether the table represents a linear or nonlinear function, we need to examine the rate of change of the function. Linear functions have a constant rate of change, meaning that the difference between consecutive [tex]$y$[/tex]-values divided by the difference between consecutive [tex]$x$[/tex]-values (slope) is always the same.
Let's analyze the provided table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
x & -2 & -1 & 0 & 1 & 2 \\
\hline
y & 4 & 2 & 0 & 2 & 4 \\
\hline
\end{tabular}
\][/tex]
### Step-by-Step Solution
1. Calculate the differences in the [tex]$x$[/tex]-values:
[tex]\[
\Delta x = x_{i+1} - x_i
\][/tex]
[tex]\[
\Delta x = [-1 - (-2), 0 - (-1), 1 - 0, 2 - 1]
\][/tex]
[tex]\[
\Delta x = [1, 1, 1, 1]
\][/tex]
2. Calculate the differences in the [tex]$y$[/tex]-values:
[tex]\[
\Delta y = y_{i+1} - y_i
\][/tex]
[tex]\[
\Delta y = [2 - 4, 0 - 2, 2 - 0, 4 - 2]
\][/tex]
[tex]\[
\Delta y = [-2, -2, 2, 2]
\][/tex]
3. Calculate the rate of change for each pair of points:
[tex]\[
\text{Rate of change} = \frac{\Delta y}{\Delta x}
\][/tex]
[tex]\[
\text{Rate of change} = \left[\frac{-2}{1}, \frac{-2}{1}, \frac{2}{1}, \frac{2}{1}\right]
\][/tex]
[tex]\[
\text{Rate of change} = [-2.0, -2.0, 2.0, 2.0]
\][/tex]
4. Check if the rate of change is constant:
[tex]\[
\text{Rate of change} = [-2.0, -2.0, 2.0, 2.0]
\][/tex]
The rates of change are not all the same. Some are -2.0 and others are 2.0.
### Conclusion
Since the rate of change of the output values ([tex]\(y\)[/tex]) is not constant, the table represents a nonlinear function.
Therefore, the correct statement is:
"The table represents a nonlinear function because there is not a constant rate of change in the output values."
