Chapter 8.2, Problem 71E

To determine

To show: whether the function is exponential growth or exponential decay. Also, sketch the graph of the function.

Answer to Problem 71E

The function represents exponential decay.

  

Explanation of Solution

Given information:

  $y=e^{-3x}$

Formula used:

An exponential function has the form $y=a{b}^x$ , where $a\ne 0$ and the base b is a positive real number other than 1.

If $a>0$ and $b>1$ , then $y=a{b}^x$ is an exponential; decay function, and b is called the decay factor.

If $a>0$ and $0<b<1$ , then $y=a{b}^x$ is an exponential; decay function, and b is called the decay factor.

Calculation:

  $\begin{array}{l}y=e^{-3x}\\ y={\left(e^{-1}\right)}^{3x}\end{array}$

Step1. Identifying the value of the base.

Since, the base is $e^{-1}=0.367879441$ , is greater than 0 and less than 1.

So, the function represents exponential decay.

Step2. Making of table with different values

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$y$	$\begin{array}{l}y=e^{-3(-2)}\\ =e^6\\ =403.4287\end{array}$	$\begin{array}{l}y=e^{-3(-1)}\\ =e^3\\ =20.085536\end{array}$	$\begin{array}{l}y=e^{-3(0)}\\ =e^0\\ =1\end{array}$	$\begin{array}{l}y=e^{-3(1)}\\ =e^{-3}\\ =0.049787\end{array}$	$\begin{array}{l}y=e^{-3(2)}\\ =e^{-6}\\ =0.002478\end{array}$	$\begin{array}{l}y=e^{-3(3)}\\ =e^{-9}\\ =0.000123\end{array}$

Step3. Plotting the points.

  

Figure1.

Step4. Start drawing the graph from right to left. A smooth curve that begins just above the x -axis, passes through the plotted points, moves up to the left.