Question

Find the average rate of change of the function f ( x ) = − 2 x 2 + 3 x − 6 , from x=0 to x=4. Note, the directions are equivalent to "Find the average rate of change over the interval [0,4]".
Average rate of change =

Answer 1

The average rate of change has a relationship with slopes. Recall that we can write the slope of a line as follows:

$slope=\frac{Change \ y}{Change \ x}$

We have the following function:

$f(x)=-2x^2+3x-6$

To find the Average Rate of Change (ARC), we need to compute the following:

$ARC=\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}=\frac{Change \ y}{Change \ x}$

So:

$x_{1}=0 \\ f(x_{1})= -2(0)^2+3(0)-6=-6 \\ \\ x_{2}=4 \\ f(x_{2})=-2(4)^2+3(4)-6=-26$

Finally, the Average Rate of Change is:

$ARC=\frac{-26-(-6)}{4-0} \\ \\ \therefore \boxed{ARC=-5}$