Question

average rate of change quick check
use the image to answer the question.
describe the graph of f(x) by selecting the correct statement.
(1 point)
the average rate of change is negative on the interval 1,3 and also on the interval 6,7.
the average rate of change is negative only on the interval 1,3.
the average rate of change is negative only on the interval 6,7.
the average rate of change is negative on the interval 0,1 and on the interval 3,4

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 given by $\frac{f(b)-f(a)}{b - a}$. If $f(b)-f(a)<0$ and $b - a>0$ (since $b>a$ for an interval $[a,b]$), the average rate of change is negative.

Step2: Analyze interval $[1,3]$

For the interval $[1,3]$, $f(1) = 1$ and $f(3)=0$. Then $\frac{f(3)-f(1)}{3 - 1}=\frac{0 - 1}{2}=-\frac{1}{2}<0$.

Step3: Analyze interval $[6,7]$

For the interval $[6,7]$, $f(6) = 2$ and $f(7)=0$. Then $\frac{f(7)-f(6)}{7 - 6}=\frac{0 - 2}{1}=- 2<0$.

Step4: Analyze other intervals

For $[0,1]$, $f(0)=-1$ and $f(1) = 1$, $\frac{f(1)-f(0)}{1-0}=\frac{1-(-1)}{1}=2>0$. For $[3,4]$, $f(3) = 0$ and $f(4)=1$, $\frac{f(4)-f(3)}{4 - 3}=\frac{1-0}{1}=1>0$.

Answer:

The average rate of change is negative on the interval $[1,3]$ and also on the interval $[6,7]$.