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Graphs of Exponential Functions

Growth and decay functions with varying compounding intervals

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Exponential Functions
License: CC BY-NC 3.0

In a laboratory, one strain of bacteria can double in number every 15 minutes. Suppose a culture starts with 60 cells, use your graphing calculator or a table of values to show the sample’s growth after 2 hours. Use the function \begin{align*}b=60 \times 2^q\end{align*}where \begin{align*}b\end{align*} is the number of cells after q quarter hours.

In this concept, you will learn to recognize, evaluate and graph exponential functions.

Exponential Functions

An exponential function is any function that can be written in the form \begin{align*}y=ab^x\end{align*} , where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are constants\begin{align*}a \neq 0,b > 0\end{align*}, and \begin{align*}b\neq 1\end{align*}.

Two girls in a small town once shared a secret, just between the two of them. They couldn’t stand it though, and each of them told three friends. Of course, their friends couldn’t keep secrets, either, and each of them told three of their friends. Those friends told three friends, and those friends told three friends, and so on... and pretty soon the whole town knew the secret. There was nobody else to tell!

These girls experienced the startling effects of an exponential function. If you start with the two girls who each told three friends, you can see that they told six people or \begin{align*}2 \times 3\end{align*}. Those six people each told three others, so that \begin{align*}6 \times 3\end{align*} or \begin{align*}2 \times 3 \times 3\end{align*}. They told 18 people. Those 18 people each told 3, so that now is \begin{align*}18 \times 3\end{align*} or \begin{align*}2 \times 3 \times 3 \times 3\end{align*} or 54 people.

You can see how this is growing and you could show the number of people told in each round of gossip with a function: \begin{align*} y=ab^x\end{align*} where \begin{align*}y \end{align*} is the number of people told, \begin{align*}a\end{align*} is the two girls who started the gossip, \begin{align*}b\end{align*} is the number of friends that they each told, and \begin{align*}x\end{align*} is the number of rounds of gossip that occurred.

You could make a table of values and calculate the number of people told after each round of gossip. Use the function \begin{align*}y=2 \times 3^x\end{align*} where \begin{align*}y\end{align*} is the number of people told and \begin{align*}x\end{align*} is the number of rounds of gossip that occurred.

\begin{align*}x\end{align*} rounds of gossip 0 1 2 3 4 5
\begin{align*}y\end{align*} people told 2 6 18 54 162 486

Next, graph the relationship between the rounds of gossip and the number of people told.

License: CC BY-NC 3.0

How can you tell if a function is an exponential function?

If your function can be written in the form \begin{align*}y=ab^x\end{align*}, where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are constants, \begin{align*} a \neq 0,b > 0\end{align*}, and \begin{align*}b \neq 1\end{align*}, then it must be exponential. In quadratic equations, your functions were always to the 2nd power. In exponential functions, the exponent is a variable. Their graphs will have a characteristic curve either upward or downward.

Let’s look at some examples of exponential functions.

  1. \begin{align*}y=2^x\end{align*}
  2. \begin{align*}c=4 \times 10^a\end{align*}
  3. \begin{align*}y=2 \times \left( \frac{2}{3} \right)^x\end{align*}
  4. \begin{align*}t=4 \times 10^y\end{align*}

Now, here are some examples that are not exponential functions

  1. \begin{align*}y=3 \times 1^x\end{align*} because \begin{align*}b = 1\end{align*}.
  2. \begin{align*}n= 0 \times 3^p\end{align*} because \begin{align*}a = 0\end{align*}.
  3. \begin{align*}y=(-4)^x\end{align*} because \begin{align*}b<0\end{align*}.
  4. \begin{align*}y=-6 \times 0^x\end{align*} because \begin{align*} b \le 1\end{align*}.

Exponential functions can be graphed by using a table of values like you did for quadratic functions. Substitute values for \begin{align*}x\end{align*} and calculate the corresponding values for \begin{align*}y\end{align*}.

Let’s look at an example.

Graph \begin{align*}y=2^x\end{align*}.

First, fill in the table of values.

\begin{align*}x \end{align*} \begin{align*}y=2^x\end{align*}   \begin{align*}y\end{align*}
\begin{align*}-3\end{align*}  \begin{align*}y=2^{-3} \end{align*}  \begin{align*}\frac{1}{8}\end{align*}
 \begin{align*}-2\end{align*}  \begin{align*}y=2^{-2}\end{align*}  \begin{align*}\frac{1}{4}\end{align*}
 \begin{align*}-1\end{align*}  \begin{align*}y=2^{-1} \end{align*}  \begin{align*}\frac{1}{2}\end{align*}
   \begin{align*}y=2^{0} \end{align*}  \begin{align*}1\end{align*}
 \begin{align*}1\end{align*}  \begin{align*}y=2^{1}\end{align*}  \begin{align*}2\end{align*}
 \begin{align*}2\end{align*}  \begin{align*}y=2^{2} \end{align*}  \begin{align*}4\end{align*}
\begin{align*}3\end{align*}  \begin{align*}y=2^{3} \end{align*}  \begin{align*}8\end{align*}

Next, graph the function.

License: CC BY-NC 3.0

Notice that the shapes of the graphs are not parabolic like the graphs of quadratic functions. Also, as the \begin{align*}x\end{align*} value gets lower and lower, the \begin{align*}y\end{align*} value approaches zero but never reaches it. As the \begin{align*}x\end{align*} value gets even smaller, the \begin{align*}y\end{align*} value may get infinitely close to zero but will never cross the \begin{align*}x\end{align*}-axis.


Example 1

Earlier, you were given a problem about the scientists and the bacteria. The scientists are studying a strain of bacteria that doubles in number every 15 minutes. The function \begin{align*}b=60 \times 2^q\end{align*} represents the growth rate of the bacteria where \begin{align*}b\end{align*} is the number of cells there are after \begin{align*}q\end{align*} quarter hours.

First, create a t-table to go with the equation of the function.

 \begin{align*}q\end{align*}  \begin{align*}b=60 \times 2^q \end{align*}  \begin{align*}b\end{align*}
0  \begin{align*}b=60 \times 2^0 \end{align*} 60
1  \begin{align*}b=60 \times 2^1\end{align*} 120
2  \begin{align*}b=60 \times 2^2 \end{align*} 240
3  \begin{align*}b=60 \times 2^3 \end{align*} 480
4  \begin{align*}b=60 \times 2^4 \end{align*} 960
5  \begin{align*}b=60 \times 2^5 \end{align*} 1920
6  \begin{align*}b=60 \times 2^6 \end{align*} 3840
7  \begin{align*}b=60 \times 2^7 \end{align*} 7680
8  \begin{align*}b=60 \times 2^8 \end{align*} 15360

Next, graph the function.

License: CC BY-NC 3.0

Example 2

Graph the following.

\begin{align*}y= 2 \times \left( \frac{2}{3} \right)^x\end{align*}

First, fill in the table of values.

 \begin{align*}x\end{align*}  \begin{align*}-3\end{align*}  \begin{align*}-2\end{align*}  \begin{align*}-1\end{align*}    \begin{align*}1\end{align*}  \begin{align*}2\end{align*}  \begin{align*}3\end{align*}
 \begin{align*}y=2 \times \left( \frac{2}{3} \right)^x\end{align*}  \begin{align*}y=2 \left( \frac{2}{3} \right)^{-3}\end{align*}  \begin{align*}y=2 \left( \frac{2}{3} \right)^{-2}\end{align*}  \begin{align*}y=2 \left( \frac{2}{3} \right)^{-1}\end{align*} \begin{align*}y=2 \left( \frac{2}{3} \right)^{0}\end{align*}  \begin{align*}y=2 \left( \frac{2}{3} \right)^{1}\end{align*}  \begin{align*}y=2 \left( \frac{2}{3} \right)^{2}\end{align*}  \begin{align*}y=2 \left( \frac{2}{3} \right)^{3}\end{align*}
 \begin{align*}y\end{align*}  \begin{align*}\frac{27}{4}\end{align*}  \begin{align*}\frac{9}{2}\end{align*}  \begin{align*}3\end{align*}  \begin{align*}2\end{align*}  \begin{align*}\frac{4}{3}\end{align*}  \begin{align*}\frac{8}{9}\end{align*}  \begin{align*}\frac{16}{27}\end{align*}

Next, graph the function.

License: CC BY-NC 3.0

Example 3

Identify the function \begin{align*}y=4^x\end{align*}.

\begin{align*}y=4^x\end{align*} is an exponential function.

Example 4

Identify the function \begin{align*}y=3x-1\end{align*}.

\begin{align*}y=3x-1\end{align*} is a linear function.

Example 5

Identify the function \begin{align*}y=ax^2-bx+c\end{align*}.

\begin{align*}y=ax^2-bx+c\end{align*} is a quadratic function.


Classify the following functions as exponential or not exponential. If it is not exponential, state the reason why.

  1. \begin{align*}y=7^x\end{align*}
  2. \begin{align*}c=-2 \times 10^d\end{align*}
  3. \begin{align*}y=1^x\end{align*}
  4. \begin{align*} y=4^x\end{align*}
  5. \begin{align*}n=0 \times \left( \frac{1}{2} \right)^x\end{align*}
  6. \begin{align*}y=5 \times \left( \frac{4}{3} \right)^x\end{align*}
  7. \begin{align*}y=(-7)^x\end{align*}
  8. Use a table of values to graph the function \begin{align*}y=3^x\end{align*}.
  9. Use a table of values to graph the function \begin{align*}y=\left( \frac{1}{3} \right)^x\end{align*}.
  10. What type of graph did you make in number 7?
  11. What type of graph did you make in number 8?
  12. Use a table of values to graph the function \begin{align*}y=-2^x\end{align*}.
  13. Use a table of values to graph the function \begin{align*}y=5^x\end{align*}.
  14. Use a table of values to graph the function \begin{align*}y=-5^x\end{align*}.
  15. Use a table of values to graph the function \begin{align*}y=6^x\end{align*} .

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.16. 

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Asymptotic A function is asymptotic to a given line if the given line is an asymptote of the function.
Exponential Function An exponential function is a function whose variable is in the exponent. The general form is y=a \cdot b^{x-h}+k.
grows without bound If a function grows without bound, it has no limit (it stretches to \infty).
Horizontal Asymptote A horizontal asymptote is a horizontal line that indicates where a function flattens out as the independent variable gets very large or very small. A function may touch or pass through a horizontal asymptote.
Transformations Transformations are used to change the graph of a parent function into the graph of a more complex function.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0
  5. [5]^ License: CC BY-NC 3.0

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