The rate of change is given by the slope of the given function, which is the coefficient of the variable x. Then, the given rate of change is
[tex]\frac{7}{4}[/tex]In order to solve this question, we need to find the slope for every case. In general, the slope formula for 2 given points is
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Lets start.
Case A.
We can choose 2 points in our table. If we choose
[tex]\begin{gathered} (x_1,y_1)=(4,2) \\ (x_2,y_2)=(8,9) \end{gathered}[/tex]by substituting these points into the slope formula, we get
[tex]\begin{gathered} m=\frac{9-2}{8-4} \\ m=\frac{7}{4} \end{gathered}[/tex]which have the same rate that our given function.
Case B.
If we choose points
[tex]\begin{gathered} (x_1,y_1)=(-4,-2) \\ (x_2,y_2)=(0,3) \end{gathered}[/tex]the slope is given by
[tex]\begin{gathered} m=\frac{3-(-2)}{0-(-4)} \\ m=\frac{5}{4} \end{gathered}[/tex]which is less than the given rate of our function
Case C.
If we choose points
[tex]\begin{gathered} (x_1,y_1)=(0,5) \\ (x_2,y_2)=(4,-2) \end{gathered}[/tex]the slope is
[tex]\begin{gathered} m=\frac{-2-5}{4-0} \\ m=-\frac{7}{4} \end{gathered}[/tex]which is less than the given rate of our function because its a negative number.
Case D.
If we choose points
[tex]\begin{gathered} (x_1,y_1)=(-2,-4) \\ (x_2,y_2)=(0,1) \end{gathered}[/tex]we get
[tex]\begin{gathered} m=\frac{1-(-4)}{0-(-2)} \\ m=\frac{5}{2} \end{gathered}[/tex]which is greater than the given rate.
Therefore, the answer is option D
