problems 24—26\nf(x)=x² + 2x\nwrite the correct number in each box in simplest form without…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$.

Step2: Solve problem 24

First, find $f(1)$ and $f(5)$. $f(1)=1^{2}+2\times1=1 + 2=3$. $f(5)=5^{2}+2\times5=25 + 10=35$. Then, calculate the average rate of change: $\frac{f(5)-f(1)}{5 - 1}=\frac{35 - 3}{4}=\frac{32}{4}=8$.

Step3: Solve problem 25

Find $f(-3)$ and $f(4)$. $f(-3)=(-3)^{2}+2\times(-3)=9-6 = 3$. $f(4)=4^{2}+2\times4=16 + 8=24$. Calculate the average rate of change: $\frac{f(4)-f(-3)}{4-(-3)}=\frac{24 - 3}{7}=\frac{21}{7}=3$.

Step4: Solve problem 26

Find $f(-4)$ and $f(2)$. $f(-4)=(-4)^{2}+2\times(-4)=16-8 = 8$. $f(2)=2^{2}+2\times2=4 + 4=8$. Calculate the average rate of change: $\frac{f(2)-f(-4)}{2-(-4)}=\frac{8 - 8}{6}=0$.

Answer:

24. 8
25. 3
26. 0