Question

average rate of change quick check
use the image to answer the question
given the graph of f(x), on which interval is the average rate of change the greatest? (1 point)
the function has the greatest average rate of change over the interval 0,1
the function has the greatest average rate of change over the interval 1,4
the function has the greatest average rate of change over the interval 4,5
the function has the greatest average rate of change over the interval 5,6

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 $\frac{f(b)-f(a)}{b - a}$, which is the slope of the secant - line connecting the points $(a,f(a))$ and $(b,f(b))$.

Step2: Calculate average rate of change for $[0,1]$

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

Step3: Calculate average rate of change for $[1,4]$

For the interval $[1,4]$, $a = 1$, $b = 4$, $f(1)=1$, $f(4)=-1$. Then $\frac{f(4)-f(1)}{4 - 1}=\frac{-1 - 1}{3}=\frac{-2}{3}$.

Step4: Calculate average rate of change for $[4,5]$

For the interval $[4,5]$, $a = 4$, $b = 5$, $f(4)=-1$, $f(5)=-1$. Then $\frac{f(5)-f(4)}{5 - 4}=\frac{-1-(-1)}{1}=0$.

Step5: Calculate average rate of change for $[5,6]$

For the interval $[5,6]$, $a = 5$, $b = 6$, $f(5)=-1$, $f(6)=-1$. Then $\frac{f(6)-f(5)}{6 - 5}=\frac{-1-(-1)}{1}=0$.

Answer:

The function has the greatest average rate of change over the interval $[0,1]$.