Chapter 9.6, Problem 43HP

To determine

An exponential function that represents a real-world situation, and graph the function

Answer to Problem 43HP

Time (hour)	Number of Bacteria$2^x$
0	1
1	2
2	4
3	8
4	16
5	32
6	64

  

Explanation of Solution

Given:

An exponential function

  $\begin{array}{c}y=a{b}^x\\ ifa=1\text{and}b=2\\ y=2^x\end{array}$

Concept Used:

  $y=a{b}^x$

In rectangular coordinate system any point be in the form of $\left(x,y\right)$ .

• In Quadrant I, both the coordinates of ordered pair are positive, that is, $\left(+,+\right)$ .
• In Quadrant II, first number is negative and second is positive, that is, $\left(-,+\right)$ .
• In Quadrant III, both the coordinates of ordered pair are negative, that is, $\left(-,-\right)$ .
• In Quadrant IV, first point is positive and second is negative, that is, $\left(+,-\right)$ .

Calculation:

In order to find the table we put the values of x in $2^x$ and let’s assume that the number of bacteria is increasing to time per hours and then simplify further as shown below:

  $\begin{array}{c}Atx=0\\ n=2^x=2^0\\ =1\\ Atx=1\\ n=2^x=2^1\\ =2\\ Atx=2\\ n=2^x=2^2\\ =4\\ Atx=3\\ n=2^x=2^3\\ =8\\ Atx=4\\ n=2^x=2^4\\ =16\\ Atx=5\\ n=2^x=2^5\\ =32\\ Atx=6\\ n=2^x=2^6\\ =64\end{array}$

Time (hour)	Number of Bacteria$2^x$
0	1
1	2
2	4
3	8
4	16
5	32
6	64

In order to plot the points $\left(0,1\right),\left(1,2\right),\left(2,4\right),\left(3,8\right),\left(4,16\right),\left(5,32\right),\text{and}\left(6,64\right)$ , observe that they all numbers are same they are positive, so the point will be in first quadrant. The graph of the point is shown below:

  

Analyze:

The number of bacteria is double to the previous one.The function that models the pattern in the table is $y=2^x$