homework assignment 3.3 rates of change and behavior of graphs score: 3/11 answered: 3/11 question 5 find…

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 1$ and $f(x)=4x^{2}-5$.

Step2: Find $f(1)$ and $f(b)$

First, find $f(1)$: Substitute $x = 1$ into $f(x)=4x^{2}-5$, we get $f(1)=4\times1^{2}-5=4 - 5=-1$. Then, find $f(b)$: Substitute $x = b$ into $f(x)=4x^{2}-5$, we get $f(b)=4b^{2}-5$.

Step3: Calculate the average rate of change

Using the formula $\frac{f(b)-f(1)}{b - 1}$, substitute $f(1)=-1$ and $f(b)=4b^{2}-5$: [ \begin{align*} \frac{f(b)-f(1)}{b - 1}&=\frac{(4b^{2}-5)-(-1)}{b - 1}\ &=\frac{4b^{2}-5 + 1}{b - 1}\ &=\frac{4b^{2}-4}{b - 1}\ &=\frac{4(b^{2}-1)}{b - 1}\ &=\frac{4(b - 1)(b + 1)}{b - 1}\ &=4(b + 1) \end{align*} ]

Answer:

$4(b + 1)$