Q. 61 - Precalculus Enhanced with Graphing Utilities

Question

Find the average rate of change of $f\left(x\right)=-2x^2+4$

(a)  From $0$ to $2$

(b) From $1$ to $3$

(c) From $1$ to $4$

Step-by-Step Solution

Answer

(a) The average rate of change is $-4$.

(b) The average rate of change is $-8.$

(c) The average rate of change is $-10.$

Step 1. Given information.

We have:

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

Part a. Step 1. Find the average rate of change from 0 to 2 .

The function is $f\left(x\right)=-2x^2+4$.

Average rate of change is given by:

$\frac{∆y}{∆x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Here, $a=0, b=2$

$\frac{∆y}{∆x}=\frac{f\left(2\right)-f\left(0\right)}{2-0}$

$$\begin{gathered} =\frac{-2\left(2^2\right)+4-(-2{\left(0\right)}^2+4)}{2-0} \\ =-4 \end{gathered}$$

Part b. Step 1. Find the average rate of change from 1 to 3 .

The function is $f\left(x\right)=-2x^2+4$.

Average rate of change is given by: 

$\frac{∆y}{∆x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Here, $a=1,b=3$

$$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(3\right)-f\left(1\right)}{3-1} \\ \frac{∆y}{∆x}=\frac{-2{\left(3\right)}^2+4-\left(-2{\left(1\right)}^2+4\right)}{2} \\ =-8 \end{gathered}$$

Part c. Step 1. Find the average rate of change from 1 to 4 .

The function is $f\left(x\right)=-2x^2+4$.

Average rate of change is given by: 

$\frac{∆y}{∆x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Here, $a=1$ and $b=4$:

$$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(4\right)-f\left(1\right)}{4-1} \\ \frac{∆y}{∆x}=\frac{-2{\left(4\right)}^2+4-(-2{\left(1\right)}^2+4)}{3} \\ =-10 \end{gathered}$$