Question

Arrange these functions from the greatest to the least value based on the average rate of change in the specified Interval. Rx) = 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

$\begin{gathered} \text{Arranging from greatest to least value based on value obtained, we have} \\ x^2+3x,3x-8,x^2-2x,x^2-5 \\ 1,2,3,4 \end{gathered}$

To find the average rate of change of a function within a given interval, we use the following formula;

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

So for the first equation, we have;

$\begin{gathered} f(x)=x^2\text{ + 3x} \\ a\text{ = -2} \\ b\text{ = 3} \\ f(a)=f(-2)=(-2)^2+3(-2)=4-6=-2 \\ f(b)=f(3)=3^2\text{ + 3(3)=9+9 = 18} \\ \\ So; \\ \frac{f(b)-f(a)}{b-a}\text{ = }\frac{18-(-2)}{3-(-2)}=\frac{18+2}{3+2}=\frac{20}{5}=4 \end{gathered}$

For the second equation, we have;

$\begin{gathered} f(x)\text{ = 3x-8} \\ a=4 \\ b=5 \\ f(a)=f(4)=3(4)-8=12-8=4 \\ f(b)=f(5)=3(5)-8=15-8=7 \\ \frac{f(b)-f(a)}{b-a}=\frac{7-4}{5-4}=\frac{3}{1}=3 \end{gathered}$

For the third equation, we have;

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

For the last equation, we have;

$\begin{gathered} f(x)\text{ = }x^2-5 \\ a=\text{ -1} \\ b=1 \\ f(a)=f(-1)=(-1)^2-5=1-5=-4 \\ f(b)=f(1)=1^2-5=1-5=-4 \\ \\ \frac{f(b)-f(a)}{b-a}=\text{ }\frac{-4-(-4)}{1-(-1)}=\frac{0}{2}=\text{ 0} \end{gathered}$

So arranging the functions from highest to lowest based on the value obtained, we have;

$x^2+3x,3x-8,x^2-2x,x^2-5$