given the function ( f(x)=x^{2}+3x - 1 ), determine the average rate of change of the function over the…

Explanation:

Step1: Recall the formula for average rate of change

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=-3) and (b = 4).

Step2: Calculate (f(-3))

Substitute (x=-3) into (f(x)=x^{2}+3x - 1): (f(-3)=(-3)^{2}+3\times(-3)-1=9-9 - 1=-1)

Step3: Calculate (f(4))

Substitute (x = 4) into (f(x)=x^{2}+3x - 1): (f(4)=4^{2}+3\times4-1=16 + 12-1=27)

Step4: Calculate the average rate of change

Using the formula (\frac{f(b)-f(a)}{b - a}), with (a=-3), (b = 4), (f(a)=-1), (f(b)=27) (\frac{f(4)-f(-3)}{4-(-3)}=\frac{27-(-1)}{4 + 3}=\frac{27 + 1}{7}=\frac{28}{7}=4)

Answer:

(4)