To find the rate of change of a function represented by a table of values, we need to follow these steps:
1. Identify the given data points:
[tex]\( x = [1, 2, 3, 4] \)[/tex]
[tex]\( y = [-8.5, -6, -3.5, -1] \)[/tex]
2. Calculate the differences between consecutive [tex]\( x \)[/tex]-values (differences_x):
[tex]\[
\Delta x_1 = x_2 - x_1 = 2 - 1 = 1 \\
\Delta x_2 = x_3 - x_2 = 3 - 2 = 1 \\
\Delta x_3 = x_4 - x_3 = 4 - 3 = 1 \\
\][/tex]
Thus, the series of differences in [tex]\( x \)[/tex]-values is: [tex]\([1, 1, 1]\)[/tex].
3. Calculate the differences between consecutive [tex]\( y \)[/tex]-values (differences_y):
[tex]\[
\Delta y_1 = y_2 - y_1 = -6 - (-8.5) = -6 + 8.5 = 2.5 \\
\Delta y_2 = y_3 - y_2 = -3.5 - (-6) = -3.5 + 6 = 2.5 \\
\Delta y_3 = y_4 - y_3 = -1 - (-3.5) = -1 + 3.5 = 2.5 \\
\][/tex]
Thus, the series of differences in [tex]\( y \)[/tex]-values is: [tex]\([2.5, 2.5, 2.5]\)[/tex].
4. Calculate the rate of change for each interval:
[tex]\[
\text{Rate of change}_1 = \frac{\Delta y_1}{\Delta x_1} = \frac{2.5}{1} = 2.5 \\
\text{Rate of change}_2 = \frac{\Delta y_2}{\Delta x_2} = \frac{2.5}{1} = 2.5 \\
\text{Rate of change}_3 = \frac{\Delta y_3}{\Delta x_3} = \frac{2.5}{1} = 2.5 \\
\][/tex]
Thus, the series of rates of change is: [tex]\([2.5, 2.5, 2.5]\)[/tex].
Since the rate of change is constant across all intervals, we can conclude that the overall rate of change of the function is [tex]\( 2.5 \)[/tex].
Therefore, the correct answer is [tex]\( 2.5 \)[/tex].
