To determine whether the table represents a linear or nonlinear function, we can calculate the rate of change (slope) between the input (x) and output (y) values.
Let's look at the differences in y-values for corresponding x-values:
-
From x = -2 to x = -1:
\( \Delta y = 2 - 4 = -2 \) (change in y is -2) -
From x = -1 to x = 0:
\( \Delta y = 0 - 2 = -2 \) (change in y is -2) -
From x = 0 to x = 1:
\( \Delta y = 2 - 0 = 2 \) (change in y is 2) -
From x = 1 to x = 2:
\( \Delta y = 4 - 2 = 2 \) (change in y is 2)
Now, we can see that the rate of change is not constant. The first two intervals have a change of -2, but the third and fourth intervals have a change of 2.
Since the rate of change is not constant, we can conclude that the function is nonlinear.
Therefore, the best explanation from the provided options is:
The table represents a nonlinear function because there is not a constant rate of change in the output values.