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 $(b,g(b))$ on the graph of $y = g(x)$.

Step2: Analyze option A ($-3\leq x\leq - 2$)

From the graph, when $x=-3$, $g(-3)\approx0$, and when $x = - 2$, $g(-2)\approx5$. The average rate of change is $\frac{g(-2)-g(-3)}{-2-(-3)}=\frac{5 - 0}{1}=5$.

Step3: Analyze option B ($-1\leq x\leq0$)

When $x=-1$, $g(-1)\approx2$, and when $x = 0$, $g(0)\approx0$. The average rate of change is $\frac{g(0)-g(-1)}{0-(-1)}=\frac{0 - 2}{1}=-2$.

Step4: Analyze option C ($1\leq x\leq2$)

When $x = 1$, $g(1)\approx1$, and when $x = 2$, $g(2)\approx - 1$. The average rate of change is $\frac{g(2)-g(1)}{2 - 1}=\frac{-1 - 1}{1}=-2$.

Step5: Analyze option D ($3\leq x\leq4$)

When $x = 3$, $g(3)\approx1$, and when $x = 4$, $g(4)\approx - 4$. The average rate of change is $\frac{g(4)-g(3)}{4 - 3}=\frac{-4 - 1}{1}=-5$.

The magnitude of the average rate of change for option B and C is $2$, for option A is $5$ and for option D is $5$. The smallest average rate of change (in magnitude) among the options is for the interval $-1\leq x\leq0$ and $1\leq x\leq2$. Since we are asked for the least average rate of change, and considering the negative values, the average rate of change on the interval $-1\leq x\leq0$ is $\frac{g(0)-g(-1)}{0 - (-1)}=\frac{0 - 2}{1}=-2$, on the interval $1\leq x\leq2$ is $\frac{g(2)-g(1)}{2 - 1}=-2$. But if we consider the non - negative value of the rate of change (taking the absolute value), we can also note that the secant line for $-1\leq x\leq0$ has a relatively flatter slope compared to the others in a non - negative sense.

Answer:

B. $-1\leq x\leq0$