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ompare 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) , 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) , 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) 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) because b(x) is increasing over the interval but d(x) is decreasing.

1 answer

To compare the estimated average rates of change for the functions \( b(x) = \sqrt{3x + 9} \) and \( d(x) = \sqrt{-3x + 9} \) over the interval \([-12, -3]\), we can use the formula for the average rate of change:

\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]

where \( a = -12 \) and \( b = -3 \).

First, we need to calculate \( b(-12) \), \( b(-3) \), \( d(-12) \), and \( d(-3) \):

For \( b(x) = \sqrt{3x + 9} \):

  1. \( b(-12) = \sqrt{3(-12) + 9} = \sqrt{-36 + 9} = \sqrt{-27} \) (undefined)
  2. \( b(-3) = \sqrt{3(-3) + 9} = \sqrt{-9 + 9} = \sqrt{0} = 0 \)

So, the average rate of change for \( b(x) \) over \([-12, -3]\) is undefined, because we cannot compare values when one of them is not defined.

For \( d(x) = \sqrt{-3x + 9} \):

  1. \( d(-12) = \sqrt{-3(-12) + 9} = \sqrt{36 + 9} = \sqrt{45} \)
  2. \( d(-3) = \sqrt{-3(-3) + 9} = \sqrt{9 + 9} = \sqrt{18} \)

Now, calculate the average rate of change for \( d \):

\[ d(-12) = \sqrt{45} \quad \text{and} \quad d(-3) = \sqrt{18} \] \[ \text{Average Rate of Change of } d(x) = \frac{\sqrt{18} - \sqrt{45}}{-3 - (-12)} = \frac{\sqrt{18} - \sqrt{45}}{9} \]

Summarizing the Results:

  • The average rate of change of \( b(x) \) is undefined.
  • The average rate of change of \( d(x) \) can be calculated but is not needed to conclude since \( b(x) \) is undefined over the interval.

Conclusion:

We can't directly compare because the average rate of change of \( b(x) \) is undefined while \( d(x) \) is calculated. We can confidently say that the estimated average rate of change of \( d(x) \) is greater than the average rate of change of \( b(x) \) simply because \( b(x) \) is not defined at all points in the interval.

However, looking at your options, including comparisons about positivity or negativity of rates doesn't apply as \( b(x) \) doesn't take valid values. A conclusion cannot be drawn about positivity versus negativity without valid comparisons.

The correct statement cannot be selected as stated without further information on the values since \( b(x) \)'s average rate cannot be determined.