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 2 ≤ x ≤ 4 goes from least to greatest.

x	2	4	6	8	10	12

h(x)=-x^{2}+4x + 26

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. For the interval $2\leq x\leq4$, $a = 2$ and $b = 4$.

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

Estimate $f(2)$ and $f(4)$ from the graph. Let's assume $f(2)\approx - 2$ and $f(4)\approx - 6$. Then $\frac{f(4)-f(2)}{4 - 2}=\frac{-6+2}{2}=-2$.

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

From the table, $g(2)=15$ and $g(4)=9$. So $\frac{g(4)-g(2)}{4 - 2}=\frac{9 - 15}{2}=-3$.

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

$h(x)=-x^{2}+4x + 26$. $h(2)=-4 + 8+26=30$, $h(4)=-16 + 16+26=26$. Then $\frac{h(4)-h(2)}{4 - 2}=\frac{26 - 30}{2}=-2$.

Answer:

The order from least to greatest of the average rates of change is $g(x),f(x),h(x)$.