Question

jane planted a marigold sapling in her garden and recorded its growth every week. the plant’s height, in inches, is modeled by the function h(x), where x is the number of weeks since jane planted the sapling.

x	h(x) = 3√(7x + 1)
1	8.49
2	11.62
3	14.07
4	16.16
5	18

what is true of the function as the x-values increase?

a. the function is increasing, and the average rate of change is increasing.

b. the function is increasing, but the average rate of change is decreasing.

c. the function is decreasing, and the average rate of change is decreasing.

d. the function is decreasing, but the average rate of change is increasing.

Explanation:

Step1: Analyze function trend

The function \( h(x) = 3\sqrt{7x + 1} \). As \( x \) increases, \( 7x + 1 \) increases, so \( \sqrt{7x + 1} \) increases, and \( 3\sqrt{7x + 1} \) increases. So the function is increasing. Eliminate C and D.

Step2: Calculate average rate of change (ARC)

ARC between \( x_1 \) and \( x_2 \) is \( \frac{h(x_2)-h(x_1)}{x_2 - x_1} \).

• ARC from \( x=0 \) to \( x=1 \): \( \frac{8.49 - 3}{1 - 0}=5.49 \)
• ARC from \( x=1 \) to \( x=2 \): \( \frac{11.62 - 8.49}{2 - 1}=3.13 \)
• ARC from \( x=2 \) to \( x=3 \): \( \frac{14.07 - 11.62}{3 - 2}=2.45 \)
• ARC from \( x=3 \) to \( x=4 \): \( \frac{16.16 - 14.07}{4 - 3}=2.09 \)
• ARC from \( x=4 \) to \( x=5 \): \( \frac{18 - 16.16}{5 - 4}=1.84 \)

The ARC is decreasing as \( x \) increases.

Answer:

B. The function is increasing, but the average rate of change is decreasing.