the table gives values for a polynomial function h at selected values of x. it is known that h is increasing…

the table gives values for a polynomial function h at selected values of x. it is known that h is increasing on the interval 0<x<4. which of the following could be true about the graph of h on the interval 0<x<4? the graph of h is concave down because the average rate of change over equal - length input - value intervals is increasing. the graph of h is concave down because the average rate of change over equal - length input - value intervals is decreasing. the graph of h is concave up because the average rate of change over equal - length input - value intervals is increasing. the graph of h is concave up because the average rate of change over equal - length input - value intervals is decreasing.

the table gives values for a polynomial function h at selected values of x. it is known that h is increasing on the interval 0<x<4. which of the following could be true about the graph of h on the interval 0<x<4? the graph of h is concave down because the average rate of change over equal - length input - value intervals is increasing. the graph of h is concave down because the average rate of change over equal - length input - value intervals is decreasing. the graph of h is concave up because the average rate of change over equal - length input - value intervals is increasing. the graph of h is concave up because the average rate of change over equal - length input - value intervals is decreasing.

Answer

Explanation:

Step1: Calculate average - rate of change

The average rate of change of a function $y = h(x)$ over the interval $[a,b]$ is $\frac{h(b)-h(a)}{b - a}$. Over the interval $[0,1]$: $\frac{h(1)-h(0)}{1-0}=\frac{4 - (-2)}{1}=6$. Over the interval $[1,2]$: $\frac{h(2)-h(1)}{2 - 1}=\frac{9 - 4}{1}=5$. Over the interval $[2,3]$: $\frac{h(3)-h(2)}{3 - 2}=\frac{11 - 9}{1}=2$. Over the interval $[3,4]$: $\frac{h(4)-h(3)}{4 - 3}=\frac{12 - 11}{1}=1$.

Step2: Analyze concavity based on average - rate of change

If the average rate of change over equal - length input - value intervals is decreasing, the function is concave down. Since $6>5>2>1$, the average rate of change over equal - length input - value intervals is decreasing.

Answer:

The graph of h is concave down because the average rate of change over equal - length input - value intervals is decreasing.