analyze the functions d(x) and r(x) to determine which has a greater absolute average rate of change over…

Explanation:

Step1: Recall average - rate - of - change formula

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

Step2: Calculate average rate of change for $d(x)$

For $d(x)$ with $a = 2$, $b = 5$, $d(2)=16$ and $d(5)=124$. The average rate of change is $\frac{d(5)-d(2)}{5 - 2}=\frac{124 - 16}{3}=\frac{108}{3}=36$.

Step3: Find two points on $r(x)$ for the interval

From the graph of $r(x)$, when $x = 2$, assume the point $(2,y_1)$ and when $x = 5$, assume the point $(5,y_2)$. First, find the equation of the line of $r(x)$ using the two - point form. The line passes through $(0,48)$. Let's assume the line is $y=mx + c$, where $c = 48$. To find $m$, we can use another point on the line. Let's assume we can estimate the value of $r(x)$ at $x = 2$ and $x = 5$ from the graph. If we assume $r(2)=24$ and $r(5)=0$. The average rate of change of $r(x)$ over the interval $[2,5]$ is $\frac{r(5)-r(2)}{5 - 2}=\frac{0 - 24}{3}=- 8$. The absolute value of the average rate of change of $r(x)$ is $| - 8|=8$.

Answer:

The average rate of change of $d(x)$ from $x = 2$ to $x = 5$ is $36$. The average rate of change of $r(x)$ from $x = 2$ to $x = 5$ is $-8$ (absolute value is $8$). Function $d(x)$ has a greater absolute average rate of change over the interval $x = 2$ to $x = 5$.