Chapter 8.2, Problem 72E

To determine

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

Answer to Problem 72E

The function represents exponential decay.

  

Explanation of Solution

Given information:

  $y=3e^{-x}$

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:

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

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

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$y$	$\begin{array}{l}y=3e^{-(-2)}\\ =3e^2\\ =22.167\end{array}$	$\begin{array}{l}y=3e^{-(-1)}\\ =3e^1\\ =8.154\end{array}$	$\begin{array}{l}y=3e^0\\ =3\end{array}$	$\begin{array}{l}y=3e^{-(1)}\\ =3e^{-1}\\ =1.1036\end{array}$	$\begin{array}{l}y=3e^{-(2)}\\ =3e^{-2}\\ =0.40600\end{array}$	$\begin{array}{l}y=3e^{-(3)}\\ =3e^{-3}\\ =0.14936\end{array}$

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.