Question

select the correct answer.
jane planted a marigold sapling in her garden and recorded its growth every week. the plants 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) = 3sqrt{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 decreasing, but the average rate of change is increasing.
c. the function is increasing, but the average rate of change is decreasing.
d. the function is decreasing, and the average rate of change is decreasing.

Explanation:

Step1: Analyze function's trend

Check \( h(x) \) values as \( x \) increases. From \( x=0 \) (3) to \( x=5 \) (18), \( h(x) \) rises. So function is increasing. Eliminate B, D.

Step2: Calculate average rate of change (ARC)

ARC between \( x=a \) and \( x=b \) is \( \frac{h(b)-h(a)}{b-a} \).

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

ARC values decrease as \( x \) increases. So function is increasing, ARC is decreasing.

Answer:

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