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 $1 \leq x \leq 3$ from least to greatest.
$f(x)$ (graph of a function), $g(x)$ (table with $x$ values -1, 0, 1, 2, 3, 4 and corresponding $g(x)$ values 18, 11, 6, 3, 2, 3), $h(x) = -x^2 - 4x + 9$

Explanation:

Answer:

First, calculate average rates of change on $1 \leq x \leq 3$:

1. For $f(x)$: from graph, $f(1)=-30$, $f(3)=20$. Average rate: $\frac{20 - (-30)}{3-1}=25$
2. For $g(x)$: $g(1)=6$, $g(3)=2$. Average rate: $\frac{2 - 6}{3-1}=-2$
3. For $h(x)=-x^2 - x + 9$: $h(1)=7$, $h(3)=-3$. Average rate: $\frac{-3 - 7}{3-1}=-5$

Order from least to greatest: $h(x)$, $g(x)$, $f(x)$