Average rate of change is calculated using the formula:
[tex]\begin{gathered} =\text{ }\frac{f(b)\text{ - f(a)}}{b\text{ - a}} \\ \text{where b = large interval, a = small interval} \\ f(b)\text{ is the value of f(x) when x = b} \\ f(a)\text{ is the value of f(x) when x = a} \end{gathered}[/tex][tex]\begin{gathered} a\text{ = -1, b = 3} \\ f(a)\text{ =f(-1)= -3}(-1)^2\text{ + 5}(-1) \\ \text{f(-1)= -3(}1)^{}\text{ - 5 = -3 - 5} \\ f(-1)\text{ = -}8 \end{gathered}[/tex][tex]\begin{gathered} f(b)=f(3)=-3(3)^2\text{ + 5(3)}_{} \\ f(3)\text{ =-3(9) + 15 = -27 + 15} \\ f(3)\text{ = }-12 \end{gathered}[/tex]Substitute the values in the average rate of change:
[tex]\begin{gathered} Average\text{ rate of change }=\text{ }\frac{-12\text{ - (-8)}}{3\text{ }-(-1)} \\ Average\text{ rate of change }=\text{ }\frac{-12\text{ + 8}}{3\text{ + 1}} \\ Average\text{ rate of change }=\text{ }\frac{-4}{4} \\ Average\text{ rate of change }=\text{ -1} \end{gathered}[/tex]