8.5: Exponential Growth Functions
In previous lessons, we have seen the variable as the base. In exponential functions, the exponent is the variable and the base is a constant.
General Form of an Exponential Function: \begin{align*}y=a (b)^x\end{align*}, where \begin{align*}a=\end{align*} initial value and
\begin{align*}b= growth \ factor\end{align*}
In exponential growth situations, the growth factor must be greater than one.
\begin{align*}b>1\end{align*}
Example: A colony of bacteria has a population of 3,000 at noon on Sunday. During the next week, the colony’s population doubles every day. What is the population of the bacteria colony at noon on Saturday?
Solution: Make a table of values and calculate the population each day.
Day | 0 (Sun) | 1 (Mon) | 2 (Tues) | 3 (Wed) | 4 (Thurs) | 5 (Fri) | 6 (Sat) |
---|---|---|---|---|---|---|---|
Population (thousands) | 3 | 6 | 12 | 24 | 48 | 96 | 192 |
To get the population of bacteria for the next day we multiply the current day’s population by 2 because it doubles every day. If we define \begin{align*}x\end{align*} as the number of days since Sunday at noon, then we can write the following: \begin{align*}P= 3 \cdot 2^x\end{align*}. This is a formula that we can use to calculate the population on any day. For instance, the population on Saturday at noon will be \begin{align*}P = 3 \cdot 2^6=3 \cdot 64 = 192\end{align*} thousand bacteria. We use \begin{align*}x=6\end{align*}, since Saturday at noon is six days after Sunday at noon.
Graphing Exponential Functions
Example: Graph the equation using a table of values \begin{align*}y=2^x\end{align*}.
Solution: Make a table of values that includes both negative and positive values of \begin{align*}x\end{align*}. Substitute these values for \begin{align*}x\end{align*} to get the value for the \begin{align*}y\end{align*} variable.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
–3 | \begin{align*}\frac{1}{8}\end{align*} |
–2 | \begin{align*}\frac{1}{4}\end{align*} |
–1 | \begin{align*}\frac{1}{2}\end{align*} |
0 | 1 |
1 | 2 |
2 | 4 |
3 | 8 |
Plot the points on the coordinate axes to get the graph below. Exponential functions always have this basic shape: They start very small and then once they start growing, they grow faster and faster, and soon they become huge.
Comparing Graphs of Exponential Functions
The shape of the exponential graph changes if the constants change. The curve can become steeper or shallower.
Earlier in the lesson, we produced a graph for \begin{align*}y=2^x\end{align*}. Let’s compare that graph with the graph of \begin{align*}y = 3 \cdot 2^x\end{align*}.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
–2 | \begin{align*}3 \cdot 2^{-2} = 3 \cdot \frac{1}{2^2} =\frac{3}{4}\end{align*} |
–1 | \begin{align*}3 \cdot 2^{-1} = 3 \cdot \frac{1}{2^1} = \frac{3}{2}\end{align*} |
0 | \begin{align*}3 \cdot 2^0 = 3\end{align*} |
1 | \begin{align*}3 \cdot 2^1 = 6\end{align*} |
2 | \begin{align*}3 \cdot 2^2 = 3 \cdot 4 = 12\end{align*} |
3 | \begin{align*}3 \cdot 2^3 = 3 \cdot 8 = 24\end{align*} |
We can see that the function \begin{align*}y=3 \cdot 2^x\end{align*} is bigger than the function \begin{align*}y=2^x\end{align*}. In both functions, the value of \begin{align*}y\end{align*} doubles every time \begin{align*}x\end{align*} increases by one. However, \begin{align*}y=3 \cdot 2^x\end{align*} starts with a value of 3, while \begin{align*}y=2^x\end{align*} starts with a value of 1, so it makes sense that \begin{align*}y=3 \cdot 2^x\end{align*} would be bigger.
Solving Real-World Problems with Exponential Growth
Example: The population of a town is estimated to increase by 15% per year. The population today is 20,000. Make a graph of the population function and find out what the population will be ten years from now.
Solution: The population is growing at a rate of 15% each year. When something grows at a percent, this is a clue to use exponential functions.
Remember, the general form of an exponential function is \begin{align*}y=a(b)^x\end{align*}, where \begin{align*}a\end{align*} is the beginning value and \begin{align*}b\end{align*} is the total growth rate. The beginning value is 20,000. Therefore, \begin{align*}a=20,000\end{align*}.
The population is keeping the original number of people and adding 15% more each year.
\begin{align*}100\%+15\%=115\%=1.15\end{align*}
Therefore, the population is growing at a rate of 115% each year. Thus, \begin{align*}b=1.15\end{align*}.
The function to represent this situation is \begin{align*}y=20,000 \ (1.15)^x\end{align*}.
Now make a table of values and graph the function.
\begin{align*}x\end{align*} | \begin{align*}y = 20 \cdot (1.15)^x\end{align*} |
---|---|
–10 | 4.9 |
–5 | 9.9 |
0 | 20 |
5 | 40.2 |
10 | 80.9 |
Notice that we used negative values of \begin{align*}x\end{align*} in our table. Does it make sense to think of negative time? In this case \begin{align*}x=-5\end{align*} represents what the population was five years ago, so it can be useful information.
The question asked in the problem was “What will be the population of the town ten years from now?” To find the population exactly, we use \begin{align*}x=10\end{align*} in the formula. We find \begin{align*}y=20,000 \cdot (1.15)^{10}=80,912\end{align*}.
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK-12 Basic Algebra: Exponential Growth Functions (7:41)
- What is the general equation for an exponential equation? What do the variables represent?
- How is an exponential growth equation different from a linear equation?
- What is true about the growth factor of an exponential equation?
- True or false? An exponential growth function has the following form: \begin{align*}f(x)=a(b)^x\end{align*}, where \begin{align*}a>1\end{align*} and \begin{align*}b<1\end{align*}?
- What is the \begin{align*}y-\end{align*}intercept of all exponential growth functions?
Graph the following exponential functions by making a table of values.
- \begin{align*}y=3^x\end{align*}
- \begin{align*}y=2^x\end{align*}
- \begin{align*}y=5 \cdot 3^x\end{align*}
- \begin{align*}y=\frac{1}{2} \cdot 4^x\end{align*}
- \begin{align*}f(x)=\frac{1}{3} \cdot 7^x\end{align*}
- \begin{align*}f(x)=2 \cdot 3^x\end{align*}
- \begin{align*}y=40 \cdot 4^x\end{align*}
- \begin{align*}y=3 \cdot 10^x\end{align*}
Solve the following problems.
- The population of a town in 2007 is 113,505 and is increasing at a rate of 1.2% per year. What will the population be in 2012?
- A set of bacteria begins with 20 and doubles every 2 hours. How many bacteria would be present 15 hours after the experiment began?
- The cost of manufactured goods is rising at the rate of inflation, about 2.3%. Suppose an item costs $12 today. How much will it cost five years from now due to inflation?
- A chain letter is sent out to 10 people telling everyone to make 10 copies of the letter and send each one to a new person. Assume that everyone who receives the letter sends it to 10 new people and that it takes a week for each cycle. How many people receive the letter in the sixth week?
- Nadia received $200 for her \begin{align*}10^{th}\end{align*} birthday. If she saves it in a bank with a 7.5% interest rate compounded yearly, how much money will she have in the bank by her \begin{align*}21^{st}\end{align*} birthday?
Mixed Review
- Suppose a letter is randomly chosen from the alphabet. What is the probability the letter chosen is \begin{align*}M, K\end{align*}, or \begin{align*}L\end{align*}?
- Evaluate \begin{align*}t^4 \cdot t^\frac{1}{2}\end{align*} when \begin{align*}t=9\end{align*}.
- Simplify \begin{align*}28-(x-16)\end{align*}.
- Graph \begin{align*}y-1=\frac{1}{3} (x+6)\end{align*}.