Question

find average rate of change from equation
question
given the function ( h(x) = x^2 + 3x - 5 ), determine the average rate of change of the function over the interval ( -7 leq x leq 1 ).

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( h(x) \) over the interval \([a, b]\) is given by \(\frac{h(b) - h(a)}{b - a}\). Here, \( a=-7 \) and \( b = 1 \).

Step2: Calculate \( h(1) \)

Substitute \( x = 1 \) into \( h(x)=x^{2}+3x - 5 \):
\( h(1)=(1)^{2}+3(1)-5=1 + 3-5=-1 \)

Step3: Calculate \( h(-7) \)

Substitute \( x=-7 \) into \( h(x)=x^{2}+3x - 5 \):
\( h(-7)=(-7)^{2}+3(-7)-5=49-21 - 5=23 \)

Step4: Calculate the average rate of change

Using the formula \(\frac{h(b)-h(a)}{b - a}\) with \( a=-7 \), \( b = 1 \), \( h(1)=-1 \) and \( h(-7)=23 \):
\(\frac{h(1)-h(-7)}{1-(-7)}=\frac{-1 - 23}{1 + 7}=\frac{-24}{8}=-3\)

Answer:

\(-3\)