A population of 10 mice grows at a rate of 300% every month. How many mice are in the population after six months?
Guidance
An exponential function has the variable in the exponent of the expression. All exponential functions have the form: \begin{align*}f(x)=a \cdot b^{x-h}+k\end{align*} , where @$\begin{align*}h\end{align*}@$ and @$\begin{align*}k\end{align*}@$ move the function in the @$\begin{align*}x\end{align*}@$ and @$\begin{align*}y\end{align*}@$ directions respectively, much like the other functions we have seen in this text. @$\begin{align*}b\end{align*}@$ is the base and @$\begin{align*}a\end{align*}@$ changes how quickly or slowly the function grows. Let’s take a look at the parent graph, @$\begin{align*}y=2^x\end{align*}@$ .
Example A
Graph @$\begin{align*}y=2^x\end{align*}@$ . Find the @$\begin{align*}y\end{align*}@$ -intercept.
Solution: Let’s start by making a table. Include some positive and negative values for @$\begin{align*}x\end{align*}@$ and zero.
@$\begin{align*}x\end{align*}@$ | @$\begin{align*}2^x\end{align*}@$ | @$\begin{align*}y\end{align*}@$ |
---|---|---|
3 | @$\begin{align*}2^3\end{align*}@$ | 8 |
2 | @$\begin{align*}2^2\end{align*}@$ | 4 |
1 | @$\begin{align*}2^1\end{align*}@$ | 2 |
0 | @$\begin{align*}2^0\end{align*}@$ | 1 |
-1 | @$\begin{align*}2^{-1}\end{align*}@$ | @$\begin{align*}\frac{1}{2}\end{align*}@$ |
-2 | @$\begin{align*}2^{-2}\end{align*}@$ | @$\begin{align*}\frac{1}{4}\end{align*}@$ |
-3 | @$\begin{align*}2^{-3}\end{align*}@$ | @$\begin{align*}\frac{1}{8}\end{align*}@$ |
This is the typical shape of an exponential growth function . The function grows “exponentially fast”. Meaning, in this case, the function grows in powers of 2. For an exponential function to be a growth function, @$\begin{align*}a > 0\end{align*}@$ and @$\begin{align*}b > 1\end{align*}@$ and @$\begin{align*}h\end{align*}@$ and @$\begin{align*}k\end{align*}@$ are both zero @$\begin{align*}(y=ab^x)\end{align*}@$ . From the table, we see that the @$\begin{align*}y\end{align*}@$ -intercept is (0, 1).
Notice that the function gets very, very close to the @$\begin{align*}x\end{align*}@$ -axis, but never touches or passes through it. Even if we chose @$\begin{align*}x=-50, \ y\end{align*}@$ would be @$\begin{align*}2^{-50}=\frac{1}{2^{50}}\end{align*}@$ , which is still not zero, but very close. In fact, the function will never reach zero, even though it will get smaller and smaller. Therefore, this function approaches the line @$\begin{align*}y=0\end{align*}@$ , but will never touch or pass through it. This type of boundary line is called an asymptote . In the case with all exponential functions, there will be a horizontal asymptote. If @$\begin{align*}k=0\end{align*}@$ , then the asymptote will be @$\begin{align*}y=0\end{align*}@$ .
Example B
Graph @$\begin{align*}y=3^{x-2}+1\end{align*}@$ . Find the @$\begin{align*}y\end{align*}@$ -intercept, asymptote, domain and range.
Solution: This is not considered a growth function because @$\begin{align*}h\end{align*}@$ and @$\begin{align*}k\end{align*}@$ are not zero. To graph something like this (without a calculator), start by graphing @$\begin{align*}y=3^x\end{align*}@$ and then shift it @$\begin{align*}h\end{align*}@$ units in the @$\begin{align*}x\end{align*}@$ -direction and @$\begin{align*}k\end{align*}@$ units in the @$\begin{align*}y\end{align*}@$ -direction.
Notice that the point (0, 1) from @$\begin{align*}y=3^x\end{align*}@$ gets shifted to the right 2 units and up one unit and is (2, 2) in the translated function, @$\begin{align*}y=3^{x-2}+1\end{align*}@$ . Therefore, the asymptote is @$\begin{align*}y=1\end{align*}@$ . To find the @$\begin{align*}y\end{align*}@$ -intercept, plug in @$\begin{align*}x=0\end{align*}@$ .
@$$\begin{align*}y=3^{0-2}+1=3^{-2}+1=1 \frac{1}{9} = 1. \overline{1}\end{align*}@$$
The domain of all exponential functions is all real numbers. The range will be everything greater than the asymptote. In this example, the range is @$\begin{align*}y > 1\end{align*}@$ .
Example C
Graph the function @$\begin{align*}y= -\frac{1}{2} \cdot 4^x\end{align*}@$ . Determine if it is an exponential growth function.
Solution: In this example, we will outline how to use the graphing calculator to graph an exponential function. First, clear out anything in Y=. Next, input the function into Y1= -(1/2)4^X and press GRAPH . Adjust your window accordingly.
This is not an exponential growth function, because it does not grow in a positive direction. By looking at the definition of a growth function, @$\begin{align*}a>0\end{align*}@$ , and it is not here.
Intro Problem Revisit
This is an example of exponential growth, so we can use the exponential form @$\begin{align*}f(x)=a \cdot b^{x-h}+k\end{align*}@$ . In this case, a = 10, the starting population; b = 300% or 3, the rate of growth; x-h = 6 the number of months, and k = 0.
@$$\begin{align*}P = 10 \cdot 3^6\\ = 10 \cdot 729 = 7290\end{align*}@$$
Therefore, the mouse population after six months is 7,290.
Guided Practice
Graph the following exponential functions. Determine if they are growth functions. Then, find the @$\begin{align*}y\end{align*}@$ -intercept, asymptote, domain and range. Use an appropriate window.
1. @$\begin{align*}y=3^{x-4}-2\end{align*}@$
2. @$\begin{align*}f(x)=(-2)^{x+5}\end{align*}@$
3. @$\begin{align*}f(x)=5^x\end{align*}@$
4. Abigail is in a singles tennis tournament. She finds out that there are eight rounds until the final match. If the tournament is single elimination, how many games will be played? How many competitors are in the tournament?
Answers
1. This is not a growth function because @$\begin{align*}h\end{align*}@$ and @$\begin{align*}k\end{align*}@$ are not zero. The @$\begin{align*}y\end{align*}@$ -intercept is @$\begin{align*}y=3^{0-4}-2=\frac{1}{81}-2=-1\frac{80}{81}\end{align*}@$ , the asymptote is at @$\begin{align*}y=-2\end{align*}@$ , the domain is all real numbers and the range is @$\begin{align*}y>-2\end{align*}@$ .
2. This is not a growth function because @$\begin{align*}h\end{align*}@$ is not zero. The @$\begin{align*}y\end{align*}@$ -intercept is @$\begin{align*}y=(-2)^{0+5}=(-2)^5=-32\end{align*}@$ , the asymptote is at @$\begin{align*}y=0\end{align*}@$ , the domain is all real numbers and the range is @$\begin{align*}y>0\end{align*}@$ .
3. This is a growth function. The @$\begin{align*}y\end{align*}@$ -intercept is @$\begin{align*}y=5^\circ =1\end{align*}@$ , the asymptote is at @$\begin{align*}y=0\end{align*}@$ , the domain is all real numbers and the range is @$\begin{align*}y>0\end{align*}@$ .
4. If there are eight rounds to single’s games, there are will be @$\begin{align*}2^8=256\end{align*}@$ competitors. In the first round, there will be 128 matches, then 64 matches, followed by 32 matches, then 16 matches, 8, 4, 2, and finally the championship game. Adding all these all together, there will be @$\begin{align*}128+64+32+16+8+4+2+1\end{align*}@$ or 255 total matches.
Explore More
Graph the following exponential functions. Find the @$\begin{align*}y\end{align*}@$ -intercept, the equation of the asymptote and the domain and range for each function.
- @$\begin{align*}y=4^x\end{align*}@$
- @$\begin{align*}y=(-1)(5)^x\end{align*}@$
- @$\begin{align*}y=3^x-2\end{align*}@$
- @$\begin{align*}y=2^x+1\end{align*}@$
- @$\begin{align*}y=6^{x+3}\end{align*}@$
- @$\begin{align*}y= -\frac{1}{4}(2)^x+3\end{align*}@$
- @$\begin{align*}y=7^{x+3}-5\end{align*}@$
- @$\begin{align*}y=-(3)^{x-4}+2\end{align*}@$
- @$\begin{align*}y=3(2)^{x+1}-5\end{align*}@$
- What is the y -intercept of @$\begin{align*}y=a^x\end{align*}@$ ? Why is that?
- What is the range of the function @$\begin{align*}y=a^{x-h}+k\end{align*}@$ ?
- March Madness is a single-game elimination tournament of 64 college basketball teams. How many games will be played until there is a champion? Include the championship game.
- In 2012, the tournament added 4 teams to make it a field of 68 and there are 4 "play-in" games at the beginning of the tournament. How many games are played now?
- An investment grows according the function @$\begin{align*}A=P(1.05)^t\end{align*}@$ where @$\begin{align*}P\end{align*}@$ represents the initial investment, @$\begin{align*}A\end{align*}@$ represents the value of the investment and @$\begin{align*}t\end{align*}@$ represents the number of years of investment. If $10,000 was the initial investment, how much would the value of the investment be after 10 years, to the nearest dollar?
- How much would the value of the investment be after 20 years, to the nearest dollar?