Question

if ( f(x)=-x^{2}+2x + 3 ), what is the average rate of change over ( 0,3 )?
a. 1
b. -1
c. 0
d. 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