if ( f(x)=-x^{2}+2x + 3 ), what is the average rate of change over ( 0,3 )?\na. 1\nb. -1\nc. 0\nd. 3

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function (y = f(x)) over the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}). Here, (a = 0) and (b=3).

Step2: Calculate (f(0)) and (f(3))

• For (x = 0): Substitute (x = 0) into (f(x)=-x^{2}+2x + 3). Then (f(0)=-(0)^{2}+2(0)+3=3).
• For (x = 3): Substitute (x = 3) into (f(x)=-x^{2}+2x + 3). Then (f(3)=-(3)^{2}+2(3)+3=-9 + 6+3=0).

Step3: Apply the average - rate - of - change formula

Substitute (f(0) = 3), (f(3)=0), (a = 0), and (b = 3) into (\frac{f(b)-f(a)}{b - a}). We get (\frac{f(3)-f(0)}{3-0}=\frac{0 - 3}{3}=\frac{-3}{3}=-1).

Answer:

b. -1