What Is an Exponential Function?

Exponential Functions

If you think of functions with exponents, you’re probably used to seeing something like this.

That’s the graph of y = x 2, and it is indeed a function with an exponent. But it’s not an exponential function.

In an exponential function , the independent variable, or x-value, is the exponent, while the base is a constant. For example, y = 2 x would be an exponential function. Here’s what that looks like.

The formula for an exponential function is y = ab x , where a and b are constants. You can see that this conforms to the basic pattern of a function, where you plug in some value of x and get out some value of y . But what are the two constants for? Why do you need two?

To illustrate this, let’s look at an example of something you might express with an exponential function. In this example, we’ll look at the popularity of cell phones.

Example Function

Whenever a new piece of technology comes out, people don’t all rush out to get it all at once. It starts with just a few people, and then gradually it catches on more and more, and then everyone’s using it.

Hey, that looks like an exponential function!

Just for example, let’s take cell phones. Back in the caveman days, also known as the 1980s, cell phones were pretty rare. Without going into the exact numbers, let’s say that in 1980, five people in your town had a cell phone.

Over the course of that year, each of those people persuaded one friend to get a phone, so then you had ten people with phones after one year. Then, each of those people persuaded a friend to get a phone, so after two years, there were 20 people with phones.

If you kept doubling the number every year, you’d get a really huge number really fast - that’s the whole point of an exponential function. Every year, the number increases by an increasing amount.

Now let’s get back to our equation for an exponential function: y = ab x .

Y is the number of people with phones, because that’s our dependent variable. X is the number of years since 1980, because that’s our independent variable.

We started with just five people with cell phones, so 5 is our starting value , the initial value of the function, represented by the constant a . In the first year, we multiplied that by 2.

In the second year, we took our number from the first year and multiplied that by 2. This gives us 5 x 2 x 2, which equals 5 times 2 squared. The result was 20 people. In the third year, each of those 20 people convinced a friend to get a phone, so we simply had to multiply by 2 again. This gave us 5 x 2 x 2 x 2, or 5 times 2 to the third power, which equals 40. You can see the pattern here: we’re adding 1 to the exponent every year, which means that we multiply 2 by itself one additional time every year. In this example, 2 represents the number repeatedly multiplied each step , the value raised to the power of x , represented by the constant b .

This is why we need two constants in the equation: one for the original value, and one for the value raised to the power of x . This can be a little bit confusing, because a lot of exponential functions start with just one thing to begin with, so a = 1. 1 times any number is that same number, so it looks like the function is just y = b x . But don’t be confused: a is still there! It’s just equal to 1.

Another Example

A common way that you’ll see exponential functions described in words is with a phrase like ‘increases or decreases by % per year.’ For example, an investment increases in value by one percent per year. If you’re calculating interest on a loan, you’d use this kind of equation.

Let’s take a look at an example problem to see how it works.

An investor buys a property in an up-and-coming area of town. As the area gets nicer, the value of the property increases. The value of the property increases by two percent per year. If the investor originally bought it for $500,000, then how much is it worth after five years?

Let’s plug this into our exponential function formula, y = ab x .

X is the number of years after the initial purchase. Y is the value of the property. These are our input and output variables.

A represents the initial value of the function. The initial value of this property is 500,000, so we’ll plug that in for a . Now, the tricky part is figuring out b .

In the first problem, b was 2, because we had twice as many cell phone users every year. In this case, the property is only worth two percent, or 0.02 more dollars, so its value is increasing more slowly. You might be tempted to plug in 0.02 for b , but just take a look and see what happens when you graph that.

You can see right away that this is not an increase in value! This gives us a function showing how much the property would be worth if every year it were valued at two percent of its value the year before. But we don’t want two percent of its value the year before; we want two percent more than its value the year before. To get that, we’d have to multiply by 1.02.

y = 500,000 1.02 x

If we determine some of the values of this function, we get:

Here’s what that looks like on a graph.

Ah, that’s better! You can’t quite see the slope getting steeper because the numbers are so big, but notice how y is increasing by a little bit more every time - first it increases by 10,000, then by 10,200, then by 10,404, and so on.

You can see that if you do the math by hand, it works out to the same values you get from the function; multiplying each year’s value by 1.02 to find the two percent increase gives you the same values for each year. So, for year five, which is what the question originally asked, the value would be $552,020.40. Our savvy investor made $52,000!

Lesson Summary

In this lesson, you learned about exponential functions. An exponential function is written in the form y = ab x .

y represents the output

a represents the initial value of the function

b represents the rate of growth

x represents the input

In an exponential function, a is multiplied by b x times to create y . The graph of an exponential function looks like a curve that starts off with a very flat slope but starts getting steeper and steeper over time.

You can use these functions to solve problems about everything from the growth of bacteria to the interest you earn on your bank account - try some on the quiz questions and see how you do!