Q. 26 - Precalculus Enhanced with Graphing Utilities

Question

Find the average rate of change from $2$ to $3$ for each function $f$. Be sure to simplify.

$f\left(x\right)=3x-4x^2$

Step-by-Step Solution

Answer

The average rate of change is $-17$.

Step 1. Find f ( 2 ) and f ( 3 )

Consider the function $f\left(x\right)=3x-4x^2$.

Now the function value for $x=2$ is given as

$$\begin{gathered} f\left(2\right)=3\times 2-4\times 2^2 \\ f\left(2\right)=6-4\times 4 \\ f\left(2\right)=6-16 \\ f\left(2\right)=-10 \end{gathered}$$

Again, the function value for $x=3$ is given as

$$\begin{gathered} f\left(3\right)=3\times 3-4\times 3^2 \\ f\left(3\right)=9-4\times 9 \\ f\left(3\right)=9-36 \\ f\left(3\right)=-27 \end{gathered}$$

Step 2. Find the average rate of change

The average of change for the function $f\left(x\right)$ from $2$ to $3$ is given as

$\begin{array}{rcl}\frac{f\left(3\right)-f\left(2\right)}{3-2}& =& \frac{-27-(-10)}{1}\\ & =& -27+10\\ & =& -17\end{array}$