Suppose you want to know if the amount of radioactivity in the rock sample from an asteroid is going to be dangerous to people in 100 years: exponential functions can help you figure that out. How on Earth, you ask? Well, exponential functions increase or decrease by a constant rate, just like the rate at which a radioactive element decays.
Exponential functions experience growth or decay which is faster than that of polynomials, and because of this property they are suitable to describe a variety of phenomena from everyday life, like the exponential growth of a virus (sounds familiar?) or the decay of radioactive elements mentioned above.
So here we look at exponential functions, what their properties are, and how to graph them.
What are exponential functions used for?
Exponential functions can be used to model things that don't take on negative values and that grow or decay very quickly. You will often see them when looking at things like the number of bacteria in a culture, or in investments that earn compound interest. These are examples of exponential growth. You can also find examples where exponential functions are used to model the decay of radioactive isotopes. These are examples of exponential decay.
Example of an Exponential Function in real-life
Let's take a look at a common real-life scenario we may encounter that can be modeled using exponential functions.
One of the most common uses for exponential functions is in looking at populations that grow very quickly. Suppose you start with 10 flies, and every week you have twice as many flies. How many flies do you have in 3 months?
Answer:
Name the function for counting flies ![]()
. The initial population of flies is 10, so ![]()
.
After 1 week you have twice as many flies, so
![]()
After 2 weeks the population has doubled again, so
![]()
After 3 weeks the population again doubles, so
![]()
Now you can see the general trend and get the formula ![]()
In 3 months how many flies are there? Notice that x is measured in weeks, so first convert 3 months to 12 weeks. Then
![]()
So in just 3 months, you have more than forty thousand flies!
What is an Exponential Function?
An exponential function is a function that has the form
![]()
where ![]()
and ![]()
are constants, ![]()
, and x is a real number. Rather than starting with the more general form, let's first look at the more simple form ![]()
(which is the case where ![]()
) to get a better idea of how exponential functions behave.
What are the characteristics of an exponential function?
The function ![]()
is an exponential function with ![]()
. It is a decreasing function, but it is concave up just like the previous example. It is an example of exponential decay.
Fig. 2: Exponential function 0 < a < 1.
What is the general equation of an exponential function?
The exponential function can be written in a more general way.
The constants B, k, and C take the graph of the basic exponential function and either shift, flip, or stretch them.
- C moves the graph up or down, which changes the position of the horizontal asymptote and the position of the y-intercept
- B flips the graph over the x-axis (if
![]()
), which makes the graph concave down, and will also affect the position of the y-intercept. It makes the graph increase/decrease more quickly (![]()
) or less quickly (![]()
). - k changes the rate of exponential growth or exponential decay. It makes the graph increase/decrease more quickly (
![]()
) or less quickly (![]()
). If ![]()
, then it is flipped over the y-axis. - the horizontal asymptote is the equation
![]()
- the y-intercept is at
![]()
- the domain is
![]()
- the range depends on both B and C. If
![]()
(concave up), then the range is ![]()
. If ![]()
(concave down), then the range is ![]()
.
Only the absolute value of k and B affect how quickly the exponential function increase or decreases, while the negative sign is only responsible for flipping over an axis. For example, if ![]()
,the function is flipped over the y-axis (because k is negative), and is also increases more quickly (because the absolute value of k is greater than 1). How fast the function is increasing or decreasing is related to its derivative. More information can be found in Derivative of the Exponential Function.
There are 8 possible combinations of signs for ![]()
and ![]()
that tell you whether an exponential function is going to be increasing or decreasing, as well as concave up or concave down.
Graph the function ![]()
, being sure to find all of the important points and the horizontal asymptote.
Answer:
1. First find the y-intercept by plugging in ![]()
.
![]()
Notice that the basic exponential function has the y intercept at ![]()
, and this function has it at ![]()
. The y-intercept can also be found by using the formula ![]()
![]()
.
2. Since the whole graph is shifted down by 3, that means the horizontal asymptote is also shifted down by 3. So the equation of the horizontal asymptote is ![]()
.
3. The value of B is 5, which won't flip the graph over the x-axis because ![]()
.
4. The value of k is -4. That will flip the graph over the y axis because ![]()
.
5. Next you can make a table of values for the function.
Notice that the values when x is negative get quite large! So if you try to graph it using a standard graphing window on a calculator it probably isn't going to give you a very good idea of what the graph looks like. Instead, graph it for
![]()
. Then the graph looks like this:
Fig. 3: final graph of the function.Identifying an exponential function from a graph
Can you tell if something is an exponential function just by looking at a graph? The short answer is not really. Because the domain of an exponential function is all real numbers, and because it is impossible to make an infinite-sized graph, you can't be absolutely certain that a graph is really an exponential graph. However, you can look at a graph with some points labeled on it and decide if it might be exponential or if it definitely isn't exponential.
Ways to look at a graph and tell if it is not an exponential function:
- If it doesn't have a horizontal asymptote, it isn't an exponential function
- If it changes concavity (is sometimes concave up and sometimes concave down) it isn't an exponential function
- If the domain doesn't include all real numbers, it isn't an exponential function
- If it isn't either always decreasing or always increasing, it isn't an exponential function.
By looking at the graph, decide if the function could be exponential, or if it is definitely not exponential.
Fig. 4: Check the graph to see if it is exponential.
Answer:
First, this graph is concave up, which means it could possibly be an exponential function. But the graph starts out decreasing and then at ![]()
it becomes increasing, so you can definitely say this is NOT the graph of an exponential function.
By looking at the graph, decide if the function could be exponential, or if it is definitely not exponential.
Fig. 5: Check to see if the graph is exponential.
Answer:
This graph is always increasing. And while it doesn't appear to have a horizontal asymptote it might be somewhere other than the graphed area. On the other hand, this graph is concave down for ![]()
, and is concave up for ![]()
, so you can definitely say it is NOT an exponential function.
By looking at the graph, decide if the function could be exponential, or if it is definitely not exponential.
Fig. 6: Check to see if the graph is exponential, StudySmarter Originals
Answer:
This function is certainly increasing, and it is always concave down. It might have a horizontal asymptote, but it is hard to tell from just the picture. But the domain on this graph doesn't include any negative x values, which means this is NOT an exponential function.
By looking at the graph, decide if the function could be exponential, or if it is definitely not exponential.
Fig. 7: Check to see if the graph is exponential.
Answer:
This graph looks like it has a horizontal asymptote at ![]()
, it is always decreasing, it is always concave up, and it looks like the domain is all real numbers. So it could potentially be an exponential function, based only on what we see.
Exponential Functions - Key takeaways
- Exponential functions are used to model things that either grow or decay quickly, but not both
- Exponential functions have the formula
![]()
where a, B, k, and C are constants, and ![]()
- Exponential functions have a horizontal asymptote at
![]()
- You can't tell just by looking at a graph that a function is definitely an exponential function
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