Question

Drag each tile to the correct box. Arrange these functions from the greatest to the least value based on the average rate of change in the specified interval. f(x) = x² + 3x interval: (-2, 3] f(x) = 3x - 8 interval: [4, 5] f(x) = x² - 2x interval: (-3,4) f(x) = x².5 interval: [-1.1)

Answer 1

To find the average rate of change, use the formula:

$A=\frac{f(b)-f(a)}{b-a}$

$\begin{gathered} f(x)=x^2+3x \\ \end{gathered}$

Interval, (a, b) = (-2, 3]

Let's find the average rate of change:

$\begin{gathered} f(a)=f(-2)=-2^2+3(-2)=4\text{ - 6 = -2} \\ \\ f(b)=f(3)=3^2+3(3)=9+9=18 \end{gathered}$

Average rate of change is:

$A=\frac{18-(-2)}{3-(-2)}=\frac{18+2}{3+2}=\frac{20}{5}=4$

$f(x)=3x\text{ - 8}$

Interval, (a, b) = [4,5]

Let's solve for f(a) and f(b):

$\begin{gathered} f(a)=f(4)=3(4)-8=12-8=4 \\ \\ f(b)=f(5)=3(5)-8=15-8=7 \end{gathered}$

Average rate of change =

$A=\frac{f(b)-f(a)}{b-a}=\frac{7-4}{5-4}=\frac{3}{1}=3$

$f\mleft(x\mright)=x^2-2x$

interval, (a,b) = (-3, 4)

Solve for f(a) and f(b)

$\begin{gathered} f(a)=f(-3)=-3^2-2(-3)=9+6=15 \\ \\ f(b)=f(4)=4^2-2(4)=16-8=8 \\ \\ A=\frac{8-15}{4-(-3)}=\frac{8-15}{4+3}=\frac{-7}{7}=-1 \end{gathered}$$f(x)=x^2(5)$

interval, (a,b) =[-1, 1)

$\begin{gathered} f(a)=f(-1)=-1^2(5)=5 \\ \\ f(b)=f(1)=1^2(5)=5 \\ \\ A=\frac{5-5}{1-(-1)}=\frac{0}{2}=0 \end{gathered}$