5. given the graph of the function ( f(x) ) shown below, find the average rate of change over each of the following intervals. express in simplest form. show the calculations you use to find your answers. (a) ( -3 leq x leq 1 ) (b) ( -1 leq x leq 8 ) (c) ( 1 leq x leq 9 ) (d) ( -6 leq x leq 2 )

Question

Accepted Answer

# Explanation:
## Step1: Recall the formula for average rate of change
The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$

## Step2: Find $f(x)$ values for part (a) $-3\leq x\leq1$
From the graph, when $x=-3$, $f(-3) = 5$ and when $x = 1$, $f(1)=9$
$$
\begin{align*}
\text{Average rate of change}&=\frac{f(1)-f(-3)}{1-(-3)}\
&=\frac{9 - 5}{1+3}\
&=\frac{4}{4}\
&=1
\end{align*}
$$

## Step3: Find $f(x)$ values for part (b) $-1\leq x\leq8$
From the graph, when $x=-1$, $f(-1)=-3$ and when $x = 8$, $f(8)=-5$
$$
\begin{align*}
\text{Average rate of change}&=\frac{f(8)-f(-1)}{8-(-1)}\
&=\frac{-5+3}{8 + 1}\
&=\frac{-2}{9}\
&=-\frac{2}{9}
\end{align*}
$$

## Step4: Find $f(x)$ values for part (c) $1\leq x\leq9$
From the graph, when $x = 1$, $f(1)=9$ and when $x=9$, $f(9)=2$
$$
\begin{align*}
\text{Average rate of change}&=\frac{f(9)-f(1)}{9-1}\
&=\frac{2-9}{8}\
&=\frac{-7}{8}
\end{align*}
$$

## Step5: Find $f(x)$ values for part (d) $-6\leq x\leq2$
From the graph, when $x=-6$, $f(-6)=5$ and when $x = 2$, $f(2)=9$
$$
\begin{align*}
\text{Average rate of change}&=\frac{f(2)-f(-6)}{2-(-6)}\
&=\frac{9-5}{2 + 6}\
&=\frac{4}{8}\
&=\frac{1}{2}
\end{align*}
$$

# Answer:
(a) $1$
(b) $-\frac{2}{9}$
(c) $-\frac{7}{8}$
(d) $\frac{1}{2}$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the formula for average rate of change

Step2: Find \(f(x)\) values for part (a) \(-3\leq x\leq1\)

Step3: Find \(f(x)\) values for part (b) \(-1\leq x\leq8\)

Step4: Find \(f(x)\) values for part (c) \(1\leq x\leq9\)

Step5: Find \(f(x)\) values for part (d) \(-6\leq x\leq2\)

Answer: