Question

if ( f(x)=-x^{2}+2 x+3 ), what is the average rate of change over ( 0,3 ) ?

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, \(a = 0\), \(b=3\), and \(f(x)=-x^{2}+2x + 3\).

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

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

Step3: Substitute into the formula

\(\frac{f(3)-f(0)}{3 - 0}=\frac{0 - 3}{3}=\frac{-3}{3}=-1\).

Answer:

b. -1