Question

Use the image to answer the question.

A curve labeled b of x and five points on the curve are plotted on a coordinate plane. The curve begins in quadrant 3 sloping upward from left to right, passes through quadrant 2, and then rises through quadrant 1.
▼Click for Long Desc
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)
, but both rates are positive.

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.

Answer 1

Compute each average rate of change on [-12, -3].

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

For d(x) = sqrt(−3x + 9): d(-12) = sqrt(45) ≈ 6.708, d(-3) = sqrt(18) ≈ 4.243, so
(d(-3) − d(-12))/9 = (4.243 − 6.708)/9 ≈ −2.465/9 ≈ −0.274 (negative).

Thus the average rate of change of b(x) is greater than that of d(x). The correct choice is: "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."