Question

constructed response question. upload a picture of your written work. you may use the formula sheet menu to access the formula for rate of change. find the average rate of change over the interval -1, 4 for the given function: $f(x)=sqrt{x + 5}-1$

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 $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$ and $b = 4$.

Step2: Calculate $f(-1)$

Substitute $x=-1$ into $f(x)=\sqrt{x + 5}-1$. Then $f(-1)=\sqrt{-1 + 5}-1=\sqrt{4}-1=2 - 1=1$.

Step3: Calculate $f(4)$

Substitute $x = 4$ into $f(x)=\sqrt{x + 5}-1$. Then $f(4)=\sqrt{4+5}-1=\sqrt{9}-1=3 - 1=2$.

Step4: Calculate the average rate of change

Using the formula $\frac{f(b)-f(a)}{b - a}$, we substitute $a=-1$, $b = 4$, $f(-1)=1$ and $f(4)=2$. So $\frac{f(4)-f(-1)}{4-(-1)}=\frac{2 - 1}{4 + 1}=\frac{1}{5}$.

Answer:

$\frac{1}{5}$