Question

The values on the x-axis range from negative 14 to 9 in increments of 1. The values on the y-axis range from negative 4 to 4 in increments of 1. The curve passes through the following plotted points: left parenthesis negative 12 comma negative 3 right parenthesis, left parenthesis negative 3 comma 0 right parenthesis, left parenthesis negative 2 comma 1.4 right parenthesis, left parenthesis 2 comma 2.5 right parenthesis, and left parenthesis 6 comma 3 right parenthesis. All values are approximate.

Compare the estimated average rate of change for the graphed cubic function b(x)=3x+9−−−−−√3
to the estimated average rate of change of the square root function d(x)=−3x+9−−−−−−−√
over the interval [−12,−3]
. Which comparison is true?

(1 point)

The estimated average rate of change of d(x)
is greater than the estimated average rate of change of b(x)
because d(x)
is positive over the interval but b(x)
is negative.

The estimated average rate of change of d(x)
is greater than the estimated average rate of change of b(x)
, but both rates are negative.

The estimated average rate of change of b(x)
is greater than the estimated average rate of change of d(x)
because b(x)
is increasing over the interval but d(x)
is decreasing.

The estimated average rate of change of b(x)
is greater than the estimated average rate of change of d(x)
, but both rates are positive.
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Answer 1

Compute the average rates on [-12, -3]:

b: (b(-3) − b(-12)) / 9 = (0 − (−3)) / 9 = 3/9 = 1/3 ≈ 0.333 (positive).

d: (d(-3) − d(-12)) / 9 = (√18 − √45) / 9 ≈ (4.243 − 6.708) / 9 ≈ −0.274 (negative).

Thus the average rate of change of b(x) is greater than that of d(x); b is increasing on the interval while d is decreasing. (Third choice.)