The average rate of change has a relationship with slopes. Recall that we can write the slope of a line as follows:
[tex] slope=\frac{Change \ y}{Change \ x}[/tex]
We have the following function:
[tex] f(x)=-2x^2+3x-6 [/tex]
To find the Average Rate of Change (ARC), we need to compute the following:
[tex] ARC=\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}=\frac{Change \ y}{Change \ x} [/tex]
So:
[tex] x_{1}=0 \\ f(x_{1})= -2(0)^2+3(0)-6=-6 \\ \\ x_{2}=4 \\ f(x_{2})=-2(4)^2+3(4)-6=-26 [/tex]
Finally, the Average Rate of Change is:
[tex]ARC=\frac{-26-(-6)}{4-0} \\ \\ \therefore \boxed{ARC=-5}[/tex]
