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Use the image to answer the question. A curve labeled m of x and seven points on the curve are plotted on a coordinate plane. T...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.
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)
Responses
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 times x is greater than the estimated average rate of change of b times x because d times x is positive over the interval but b times x is 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 times x is greater than the estimated average rate of change of d times x, but both rates are positive.
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 d times x is greater than the estimated average rate of change of b times 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 times x is greater than the estimated average rate of change of d times x because b times x is increasing over the interval but d times x is decreasing.
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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.
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)
Responses
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 times x is greater than the estimated average rate of change of b times x because d times x is positive over the interval but b times x is 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 times x is greater than the estimated average rate of change of d times x, but both rates are positive.
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 d times x is greater than the estimated average rate of change of b times 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 times x is greater than the estimated average rate of change of d times x because b times x is increasing over the interval but d times x is decreasing.
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Answers
There are no human answers yet.
Answered by
GPT-4o mini
To compare the estimated average rate of change of the functions \( b(x) = \sqrt[3]{3x + 9} \) and \( d(x) = \sqrt{-3x + 9} \) over the interval \([-12, -3]\), we need to understand the behavior of each function over that interval.
1. **Calculate the average rate of change for \( b(x) \):**
- Evaluate \( b(-12) \) and \( b(-3) \):
- \( b(-12) = \sqrt[3]{3(-12) + 9} = \sqrt[3]{-36 + 9} = \sqrt[3]{-27} = -3 \)
- \( b(-3) = \sqrt[3]{3(-3) + 9} = \sqrt[3]{-9 + 9} = \sqrt[3]{0} = 0 \)
- Calculate the average rate of change:
\[
\text{Average rate of change of } b(x) = \frac{b(-3) - b(-12)}{-3 - (-12)} = \frac{0 - (-3)}{9} = \frac{3}{9} = \frac{1}{3}
\]
2. **Calculate the average rate of change for \( d(x) \):**
- Evaluate \( d(-12) \) and \( d(-3) \):
- \( d(-12) = \sqrt{-3(-12) + 9} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \)
- \( d(-3) = \sqrt{-3(-3) + 9} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \)
- Calculate the average rate of change:
\[
\text{Average rate of change of } d(x) = \frac{d(-3) - d(-12)}{-3 - (-12)} = \frac{3\sqrt{2} - 3\sqrt{5}}{9}
\]
3. **Behavior of the functions in the interval:**
- \( b(x) \) is an increasing function from \( b(-12) = -3 \) to \( b(-3) = 0 \).
- \( d(x) \) is decreasing since \( d(-12) > d(-3) \) (since \( \sqrt{45} > \sqrt{18} \)).
Based on this comparison:
- The average rate of change of \( b(x) \) is positive \((\frac{1}{3})\), while the average rate of change of \( d(x) \) is negative (because \( 3\sqrt{2} < 3\sqrt{5} \)).
The applicable statement from your options is:
**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.**
1. **Calculate the average rate of change for \( b(x) \):**
- Evaluate \( b(-12) \) and \( b(-3) \):
- \( b(-12) = \sqrt[3]{3(-12) + 9} = \sqrt[3]{-36 + 9} = \sqrt[3]{-27} = -3 \)
- \( b(-3) = \sqrt[3]{3(-3) + 9} = \sqrt[3]{-9 + 9} = \sqrt[3]{0} = 0 \)
- Calculate the average rate of change:
\[
\text{Average rate of change of } b(x) = \frac{b(-3) - b(-12)}{-3 - (-12)} = \frac{0 - (-3)}{9} = \frac{3}{9} = \frac{1}{3}
\]
2. **Calculate the average rate of change for \( d(x) \):**
- Evaluate \( d(-12) \) and \( d(-3) \):
- \( d(-12) = \sqrt{-3(-12) + 9} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \)
- \( d(-3) = \sqrt{-3(-3) + 9} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \)
- Calculate the average rate of change:
\[
\text{Average rate of change of } d(x) = \frac{d(-3) - d(-12)}{-3 - (-12)} = \frac{3\sqrt{2} - 3\sqrt{5}}{9}
\]
3. **Behavior of the functions in the interval:**
- \( b(x) \) is an increasing function from \( b(-12) = -3 \) to \( b(-3) = 0 \).
- \( d(x) \) is decreasing since \( d(-12) > d(-3) \) (since \( \sqrt{45} > \sqrt{18} \)).
Based on this comparison:
- The average rate of change of \( b(x) \) is positive \((\frac{1}{3})\), while the average rate of change of \( d(x) \) is negative (because \( 3\sqrt{2} < 3\sqrt{5} \)).
The applicable statement from your options is:
**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.**
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