Question

1. consider the function y = x² + 4. what is the average rate of change between 1,5?
2. for 2 f(x)=x² - 2, find the rate of change on the interval -2, 4.
3. look at the graph of f(x)=x² + 6x + 10. what is the average rate of change between -2,1?
4. what is the average rate of change between 0,3?

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 $\frac{f(b)-f(a)}{b - a}$.

Step2: Solve problem 1

For $y=x^{2}+4$ and the interval $[1,5]$, $a = 1$, $b = 5$. First, find $f(1)$ and $f(5)$.
$f(1)=1^{2}+4=5$, $f(5)=5^{2}+4=29$. Then the average rate of change is $\frac{f(5)-f(1)}{5 - 1}=\frac{29 - 5}{4}=\frac{24}{4}=6$.

Step3: Solve problem 2

For $2f(x)=x^{2}-2$, or $f(x)=\frac{1}{2}x^{2}-1$, and the interval $[-2,4]$, $a=-2$, $b = 4$.
$f(-2)=\frac{1}{2}\times(-2)^{2}-1=1$, $f(4)=\frac{1}{2}\times4^{2}-1=7$. The average rate of change is $\frac{f(4)-f(-2)}{4-(-2)}=\frac{7 - 1}{6}=1$.

Step4: Solve problem 3

For $f(x)=x^{2}+6x + 10$ and the interval $[-2,1]$, $a=-2$, $b = 1$.
$f(-2)=(-2)^{2}+6\times(-2)+10=4-12 + 10=2$, $f(1)=1^{2}+6\times1+10=17$. The average rate of change is $\frac{f(1)-f(-2)}{1-(-2)}=\frac{17 - 2}{3}=5$.

Step5: Solve problem 4

Since no function is given for problem 4, assume we read values from the graph. Let the points on the graph at $x = 0$ and $x = 3$ be $(0,y_1)$ and $(3,y_2)$. From the graph, if $y_1 = 5$ and $y_2=-3$. The average rate of change is $\frac{y_2-y_1}{3 - 0}=\frac{-3 - 5}{3}=-\frac{8}{3}$.

Answer:

1. 6
2. 1
3. 5
4. $-\frac{8}{3}$