Question

find the average rate of change of $h(x) = -x^2 + 5x + 2$ from $x = 3$ to $x = 6$. simplify your answer as much as possible.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( h(x) \) from \( x = a \) to \( x = b \) is given by \( \frac{h(b)-h(a)}{b - a} \). Here, \( a = 3 \) and \( b = 6 \).

Step2: Calculate \( h(3) \)

Substitute \( x = 3 \) into \( h(x)=-x^{2}+5x + 2 \):
\( h(3)=-(3)^{2}+5(3)+2=-9 + 15+2=8 \)

Step3: Calculate \( h(6) \)

Substitute \( x = 6 \) into \( h(x)=-x^{2}+5x + 2 \):
\( h(6)=-(6)^{2}+5(6)+2=-36+30 + 2=-4 \)

Step4: Compute the average rate of change

Using the formula \( \frac{h(6)-h(3)}{6 - 3} \), substitute \( h(6)=-4 \) and \( h(3)=8 \):
\( \frac{-4-8}{6 - 3}=\frac{-12}{3}=-4 \)

Answer:

\(-4\)