question the functions f(x), g(x), and h(x) are shown below. select the option that represents the ordering…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = k(x)$ on the interval $[a,b]$ is $\frac{k(b)-k(a)}{b - a}$.

Step2: Calculate average rate of change of $f(x)$

From the graph of $f(x)$, when $x = 1$, assume $f(1)=y_1$ and when $x = 4$, assume $f(4)=y_2$. By counting the grid - points, if we assume each grid - square has side - length 1, we find the change in $y$ and change in $x$. Let's say $f(1)\approx10$ and $f(4)\approx30$. Then the average rate of change of $f(x)$ on $[1,4]$ is $\frac{f(4)-f(1)}{4 - 1}=\frac{30 - 10}{3}=\frac{20}{3}\approx6.67$.

Step3: Calculate average rate of change of $g(x)$

We know that $g(x)$ has values $g(1) = 14$ and $g(4)=8$. Using the average rate of change formula $\frac{g(4)-g(1)}{4 - 1}=\frac{8 - 14}{3}=\frac{-6}{3}=-2$.

Step4: Calculate average rate of change of $h(x)$

Given $h(x)=-x^{2}+2x + 15$. First, find $h(1)$ and $h(4)$. $h(1)=-1^{2}+2\times1 + 15=-1 + 2+15 = 16$. $h(4)=-4^{2}+2\times4 + 15=-16 + 8+15 = 7$. The average rate of change of $h(x)$ on $[1,4]$ is $\frac{h(4)-h(1)}{4 - 1}=\frac{7 - 16}{3}=\frac{-9}{3}=-3$.

Step5: Order the average rates of change

Comparing the average rates of change: $-3<-2<\frac{20}{3}$, so $h(x),g(x),f(x)$.

Answer:

$h(x),g(x),f(x)$