over which interval does ( f(t) ) have a negative average rate of change?\nchoose 1 answer:\n(a) ( -9,-8…

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y = f(t)) over the interval ([a,b]) is given by (\frac{f(b)-f(a)}{b - a}). For the average rate of change to be negative, (f(b)-f(a)<0) (since (b - a>0) when (b>a)). In terms of the graph, if (b>a), the function (y = f(t)) has a negative average rate of change over ([a,b]) when (f(b)<f(a)) (the function is decreasing from (t=a) to (t = b)).

Step2: Analyze option A: ([-9,-8])

For the interval ([-9,-8]), (b=-8), (a = - 9). The function is constant over ([-9,-8]) (horizontal line). Using the formula (\frac{f(-8)-f(-9)}{-8-(-9)}=\frac{f(-8)-f(-9)}{1}). Since (f(-8)=f(-9)), the average rate of change is (0).

Step3: Analyze option B: ([2,4])

For the interval ([2,4]), (b = 4), (a=2). From the graph, (f(4)=0) and (f(2)=3). Then (\frac{f(4)-f(2)}{4 - 2}=\frac{0 - 3}{2}=-\frac{3}{2}<0).

Step4: Analyze option C: ([-8,-2])

For the interval ([-8,-2]), (b=-2), (a=-8). From the graph, (f(-2)>f(-8)). Using the formula (\frac{f(-2)-f(-8)}{-2-(-8)}=\frac{f(-2)-f(-8)}{6}>0) (since (f(-2)-f(-8)>0)).

Step5: Analyze option D: ([-5,-1])

For the interval ([-5,-1]), (b=-1), (a=-5). From the graph, (f(-1)>f(-5)). Using the formula (\frac{f(-1)-f(-5)}{-1-(-5)}=\frac{f(-1)-f(-5)}{4}>0) (since (f(-1)-f(-5)>0)).

Answer:

B. ([2,4])