12.7: Exponential Functions
Introduction
The Laboratory Dilemma
“We have been given a dilemma by my friend Professor Smith,” Mr. Travis said upon the class’ return to the classroom.
“What is it?” Janet asked.
“Here we go, see what you can do with this,” Mr. Travis wrote the following problem on the board.
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 \cdot 2^q\end{align*} where \begin{align*}b\end{align*} is the number of cells there are after \begin{align*}q\end{align*} quarter hours.
To work on this problem, you have to understand exponential functions. Pay close attention during this lesson and you will know how to solve it by the end of the lesson.
What You Will Learn
In this lesson, you will learn how to complete the following skills.
- Recognize an exponential function as an equation in two variables that can be written in the form \begin{align*}y = ab^x\end{align*}.
- Evaluate and graph exponential functions using tables.
- Distinguish between exponential growth and decay.
- Evaluate and analyze real – world situations modeled by exponential functions, using a graphing calculator.
Teaching Time
I. Recognize an Exponential Function as an Equation in Two Variables that Can Be Written in the Form \begin{align*}\underline{y = ab^x}\end{align*}
Let’s think about exponential functions by looking at the following example.
Example
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 son 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 \cdot 3\end{align*}. Those six people each told three others, so that \begin{align*}6 \cdot 3\end{align*} or \begin{align*}2 \cdot 3 \cdot 3\end{align*}—they told 18 people. Those 18 people each told 3, so that now is \begin{align*}18 \cdot 3\end{align*} or \begin{align*}2 \cdot 3 \cdot 3 \cdot 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. This is called an exponential function—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 \ne 0,b>0\end{align*}, and \begin{align*}b \ne 1\end{align*}.
As we did with linear and quadratic functions, we 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 \cdot 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 \ \text{rounds of gossip} \ \quad 0 \quad 1 \quad 2 \quad \ \ 3 \quad \quad 4 \quad \quad 5\\ & y \ \text{people told} \qquad \quad \ \ 2 \quad 6 \quad 18 \quad 54 \quad 162 \quad 486\end{align*}
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 \ne 0, b>0,\end{align*} and \begin{align*}b \ne 1\end{align*}, then it must be exponential. In quadratic equations, your functions were always to the \begin{align*}2^{nd}\end{align*} power. In exponential functions, the exponent is a variable. Their graphs will have a characteristic curve either upward or downward.
Examples of Exponential Functions:
- \begin{align*}y=2^x\end{align*}
- \begin{align*}c=4 \cdot 10^d\end{align*}
- \begin{align*}y=2 \cdot \left(\frac{2}{3}\right)^x\end{align*}
- \begin{align*}t=4 \cdot 10^u\end{align*}
Not Exponential Functions:
\begin{align*} & 1.\ y=3 \cdot 1^x && 2.\ n=0 \cdot 3^p && 3. \ y=(-4)^x && 4. \ y=-6 \cdot 0^x\\ \text{because} & \quad \ b=1 && \quad \ a=0 && \quad \ b<0 && \quad \ b \le 1\end{align*}
II. Evaluate and Graph Exponential Functions Using Tables
As you saw in the gossip example, exponential functions can be graphed by using a table of values like we did for quadratic functions. Substitute values for \begin{align*}x\end{align*} and calculate the corresponding values for \begin{align*}y\end{align*}.
Example
Graph \begin{align*}y=2^x\end{align*}.
Here is the table.
\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*} |
-2 | \begin{align*}y=2^{-2}\end{align*} | \begin{align*} \frac{1}{4}\end{align*} |
-1 | \begin{align*}y=2^{-1}\end{align*} | \begin{align*} \frac{1}{2}\end{align*} |
0 | \begin{align*}y=2^0\end{align*} | 0 |
1 | \begin{align*}y=2^1\end{align*} | 2 |
2 | \begin{align*}y=2^2\end{align*} | 4 |
3 | \begin{align*}y=2^3\end{align*} | 8 |
4 | \begin{align*}y=2^4\end{align*} | 16 |
5 | \begin{align*}y=2^5\end{align*} | 32 |
6 | \begin{align*}y=2^6\end{align*} | 64 |
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.
What happens when the \begin{align*}b\end{align*} value is less than 1?
Example
Graph \begin{align*}y=2 \cdot \left( \frac{2}{3}\right)^x\end{align*}
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
\begin{align*}-3\end{align*} | \begin{align*} \frac{27}{4}\end{align*} |
-2 | \begin{align*} \frac{9}{2}\end{align*} |
-1 | 3 |
0 | 2 |
1 | \begin{align*} \frac{4}{3}\end{align*} |
2 | \begin{align*} \frac{8}{9}\end{align*} |
3 | \begin{align*} \frac{16}{27}\end{align*} |
4 | \begin{align*} \frac{32}{81}\end{align*} |
5 | \begin{align*} \frac{64}{243}\end{align*} |
6 | \begin{align*} \frac{128}{729}\end{align*} |
Do you notice a difference in the shape of the graph? The relationship in the first two examples was direct—as the \begin{align*}x\end{align*} value increased, the \begin{align*}y\end{align*} value increased. In this case, as the \begin{align*}x\end{align*} value increases, the \begin{align*}y\end{align*} value decreases. This is an inverse relationship. This brings us to our next topic.
III. Distinguish between Exponential Growth and Decay
In the last examples, you saw different kinds of graphs. In some cases with exponential functions, as the \begin{align*}x\end{align*} value increased, the \begin{align*}y\end{align*} value increased, too. This was a direct relationship known as exponential growth. As the \begin{align*}x\end{align*} value increases, the \begin{align*}y\end{align*} value grows at a very fast rate! The other graph you saw showed the opposite—as the \begin{align*}x\end{align*} value increased, the \begin{align*}y\end{align*} value decreased. This relationship is an inverse relationship known as decay. The graphs of these functions are opposites, reflected on the \begin{align*}y\end{align*}-axis.
We can also analyze growth and decay functions in real – life situations.
IV. Evaluate and Analyze Real – World Situations Modeled by Exponential Functions, Using a Graphing Calculator
A famous story tells about a courtier who presented a Persian king with a beautiful handmade chessboard. The king asked him what he would like in return for his gift and the courtier surprised the king by asking him for one grain of rice on the first square of the chessboard, two grains of rice on the second, four grains on the third, etc. The king agreed and ordered for the rice to be brought. By the \begin{align*}21^{st}\end{align*} square, over a million grains of rice were required and, by the \begin{align*}41^{st}\end{align*} square, over a quadrillion grains of rice were needed. There was simply not enough rice in all the world for the final squares.
This story reminds us of the drastic increases that we can see in exponential functions. Although this story is a fable, there are many instances in the real-world where exponential growth can be seen.
Take the human population, for example. It took hundreds of thousands of years for the population to reach 1 billion in 1804. Look at the chart below to see how increasingly quickly the next billion was reached.
Population | 1 billion | 2 billion | 3 billion | 4 billion | 5 billion | 6 billion | 7 billion | 8 billion | 9 billion |
---|---|---|---|---|---|---|---|---|---|
Year | 1804 | 1927 | 1961 | 1974 | 1987 | 1999 | 2011 | 2024 | 2042 |
Years until next billion | 123 | 34 | 13 | 13 | 12 | 12 | 13 | 18 |
Of course, many of these functions can be graphed on your graphing calculator or using a table of values and graph paper.
Example
A condominium complex charges $185 per month for the homeowners’ association fee. The rates can rise every year because of inflation but they promise not to raise the rates more than 10% each year. Keep in mind, though, that if they raise the rate by 10% the first year, the second year is now more expensive. If they raise the maximum again, they are increasing the original $185 plus the first year’s adjustment by 10%. Graph the situation for 10 years. How much could the homeowners’ fee be in ten years? Use the function \begin{align*}f=185 \cdot 1.1^t\end{align*} where \begin{align*}f\end{align*} is the fee after \begin{align*}t\end{align*} years.
\begin{align*}t\end{align*} | \begin{align*}f\end{align*} |
---|---|
0 | 185 |
1 | 203.5 |
2 | 223.85 |
3 | 246.24 |
4 | 270.86 |
5 | 297.94 |
6 | 327.74 |
7 | 360.51 |
8 | 396.56 |
9 | 436.22 |
10 | 479.84 |
Now let’s see how we can apply our work to the problem from the introduction.
Real-Life Example Completed
The Laboratory Dilemma
Here is the problem from the introduction. Reread it and then solve it for the solution.
“We have been given a dilemma by my friend Professor Smith,” Mr. Travis said upon the class’ return to the classroom.
“What is it?” Janet asked.
“Here we go, see what you can do with this,” Mr. Travis wrote the following problem on the board.
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 \cdot 2^q\end{align*} where \begin{align*}b\end{align*} is the number of cells there are after \begin{align*}q\end{align*} quarter hours.
Now solve the problem for the solution.
Solution to Real – Life Example
First, we can create a t-table to go with the equation of the function. Here are the values in that table.
\begin{align*}q\end{align*} | \begin{align*}b\end{align*} |
---|---|
0 | 60 |
1 | 120 |
2 | 240 |
3 | 480 |
4 | 960 |
5 | 1920 |
6 | 3840 |
7 | 7680 |
8 | 15360 |
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Exponential Functions
- results that expand exponentially. The graph curves upward or downward.
- Exponential Growth Graph
- a direct relationship graph each variable increases.
- Decay Graph
- an indirect relationship graph, one variable increases as the other one decreases.
Time to Practice
Directions: Classify the following functions as exponential or not exponential. If it is not exponential, state the reason why.
- \begin{align*}y=7^x\end{align*}
- \begin{align*}c=-2 \cdot 10^d\end{align*}
- \begin{align*}y=1^x\end{align*}
- \begin{align*}n=0 \cdot \left(\frac{1}{2}\right)^x\end{align*}
- \begin{align*}y=5 \cdot \left(\frac{4}{3}\right)^x\end{align*}
- \begin{align*}y=(-7)^x\end{align*}
- Use a table of values to graph the function \begin{align*}y=3^x\end{align*}. There are two parts to this answer.
- Use a table of values to graph the function \begin{align*}y=\left(\frac{1}{3}\right)^x\end{align*}. There are two parts to this answer.
- What type of graph did you make in number 7?
- What type of graph did you make in number 8?
- You invest $1000 in a savings account that pays 6% interest per year. Use your graphing calculator to show your balance over 10 years. Use the function \begin{align*}B=1000 \cdot 1.06^t\end{align*} where \begin{align*}B\end{align*} is the balance in your account and \begin{align*}t\end{align*} is the number of years you have invested. Round to the nearest whole dollar.
- Graph this function.
- In a laboratory, one strain of bacteria can double in number every 12 minutes. Suppose a culture starts with 50 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=50 \cdot 2^q\end{align*} where \begin{align*}b\end{align*} is the number of cells there are after \begin{align*}q\end{align*} segments of 12 minutes.
- Graph this function.
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