Question

The functions f(x), g(x), and h(x) are shown below. Select the option that represents the ordering of the functions according to their average rates of change on the interval -1≤x≤4 goes from least to greatest.

Answer 1

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Write the formula for the rate of change

`\begin{gathered} rate\text{ of change}=(f(b)-f(a))/(b-a) \\ rate\text{ of change}=(y_2-y_1)/(x_2-x_1) \end{gathered}`

STEP 2: Write the given intervals

`-1\leq x\leq4`

STEP 3: Find the average rate of change of f(x)

`\begin{gathered} Picking\text{ two points on the graphs, we have:} \\ (x_1,y_1)=\left(-1,5\right) \\ (x_2,y_2)=(4,0) \end{gathered}`

We substitute the coordinates into the rate of change formula:

`rate\text{ of change}=(0-5)/(4-(-1))=-(5)/(5)=-1`

STEP 4: Find the rate of change of g(x)

`\begin{gathered} (x_1,y_1)=(-1,17) \\ (x_2,y_2)=(4,2) \\ rate\text{ of change}=(2-17)/(4-(-1))=(-15)/(5)=-3 \end{gathered}`

STEP 5: Find the rate of change of h(x)

`\begin{gathered} h(x)=-x^2-5x+37 \\ x_1=-1 \\ h(-1)=-(-1^2)-5(-1)+37=-1+5+37=41 \\ x_2=4 \\ h(4)=-(4^2)-5(4)+37=-16-20+37=1 \\ The\text{ new points become:} \\ (x_1,y_1)=(-1,41) \\ (x_2,y_2)=(4,1) \\ Average\text{ rate of change:} \\ (1-41)/(4-(-1))=(-40)/(5)=-8 \end{gathered}`

STEP 6: Write the average rates of change for the functions

`\begin{gathered} f(x)=-1 \\ g(x)=-3 \\ h(x)=-8 \end{gathered}`

The average rates of change in ascending order will be -8,-3,-1

Hence, the arrangement of the functions according to their ascending order of average rates of changes are:

`h(x),g(x),f(x)`