a function is graphed below. on which interval of x is the average rate of change of the function the…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[x_1,x_2]$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$.

Step2: Calculate average rate of change for $[6,13]$

Let $(x_1,y_1)=(6,8)$ and $(x_2,y_2)=(13,28)$. Then $\frac{y_2 - y_1}{x_2 - x_1}=\frac{28 - 8}{13 - 6}=\frac{20}{7}\approx2.86$.

Step3: Calculate average rate of change for $[13,23]$

Let $(x_1,y_1)=(13,28)$ and $(x_2,y_2)=(23,34)$. Then $\frac{y_2 - y_1}{x_2 - x_1}=\frac{34 - 28}{23 - 13}=\frac{6}{10}=0.6$.

Step4: Calculate average rate of change for $[23,37]$

Let $(x_1,y_1)=(23,34)$ and $(x_2,y_2)=(37,47)$. Then $\frac{y_2 - y_1}{x_2 - x_1}=\frac{47 - 34}{37 - 23}=\frac{13}{14}\approx0.93$.

Step5: Calculate average rate of change for $[37,49]$

Let $(x_1,y_1)=(37,47)$ and $(x_2,y_2)=(49,62)$. Then $\frac{y_2 - y_1}{x_2 - x_1}=\frac{62 - 47}{49 - 37}=\frac{15}{12}=1.25$.

Answer:

The interval $[6,13]$ has the greatest average rate of change.