Chapter 8.2, Problem 73E

To determine

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

Answer to Problem 73E

The function represents exponential growth.

  

Explanation of Solution

Given information:

  $y=e^{0.25x}$

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:

  $y=e^{0.25x}$

Step1. Identifying the value of the base.

Since, the base is $e=2.718281828$ , is greater than 1.

So, the function represents exponential growth.

Step2. Making of table with different values

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$y$	$\begin{array}{l}y=e^{0.25(-2)}\\ =e^{-0.5}\\ =0.606\end{array}$	$\begin{array}{l}y=e^{0.25(-1)}\\ =e^{-0.25}\\ =0.7788\end{array}$	$\begin{array}{l}y=e^{0.25(0)}\\ =e^0\\ =1\end{array}$	$\begin{array}{l}y=e^{0.25(1)}\\ =e^{0.25}\\ =1.284\end{array}$	$\begin{array}{l}y=e^{0.25(2)}\\ =e^{0.5}\\ =1.6487\end{array}$	$\begin{array}{l}y=e^{0.25(3)}\\ =e^{0.75}\\ =2.117\end{array}$

Step3. Plotting the points.

  

Figure1.

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