Question

the graph of the function y = g(x) is given. of the following, on which interval is the average rate of change of g least?
a -3 ≤ x ≤ -2
b -1 ≤ x ≤ 0
c 1 ≤ x ≤ 2
d 3 ≤ x ≤ 4

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = g(x)$ on the interval $[a,b]$ is $\frac{g(b)-g(a)}{b - a}$, which is equivalent to the slope of the secant line connecting the points $(a,g(a))$ and $(b,g(b))$.

Step2: Analyze option A

For the interval $-3\leq x\leq - 2$, from the graph, when $x=-3$, $g(-3) = 0$, when $x = - 2$, $g(-2)=2$. The average rate of change is $\frac{g(-2)-g(-3)}{-2-(-3)}=\frac{2 - 0}{-2 + 3}=2$.

Step3: Analyze option B

For the interval $-1\leq x\leq0$, when $x=-1$, $g(-1)=1$, when $x = 0$, $g(0)=0$. The average rate of change is $\frac{g(0)-g(-1)}{0-(-1)}=\frac{0 - 1}{0 + 1}=-1$.

Step4: Analyze option C

For the interval $1\leq x\leq2$, when $x = 1$, $g(1)=0$, when $x = 2$, $g(2)=1$. The average rate of change is $\frac{g(2)-g(1)}{2 - 1}=\frac{1-0}{2 - 1}=1$.

Step5: Analyze option D

For the interval $3\leq x\leq4$, when $x = 3$, $g(3)=2$, when $x = 4$, $g(4)=-4$. The average rate of change is $\frac{g(4)-g(3)}{4 - 3}=\frac{-4 - 2}{4 - 3}=-6$.

Answer:

D. $3\leq x\leq4$