# Writing & Graphing Exponential Functions

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will explore exponential functions and their properties. Through real world examples and explanation, we will see how to write and graph these types of functions.

## Exponential Functions

Suppose someone is going to sell you their couch, and they give you two payment options. They say you can either give them \$200 right now, or you can give them a penny today, two pennies tomorrow, four pennies the next day, and continue this pattern of doubling the previous day's amount of pennies for 20 days. Which payment do you think you would agree to?

You may think the penny option sounds like you will pay much less than \$200. After all, it's just pennies! However, you may be surprised to know that though this option appears to grow slowly at first, the amount paid each day will quickly become very large. This is a characteristic of an exponential function. Exponential functions are functions that have a variable in the exponent. The amount of pennies you would have to pay them on day x can be modeled by the exponential function f(x) = 2 x.

Suppose you decided to go with the penny option, because it sounded like a better deal. By day 20, you would have to pay them f(20) = 2 20 = 1,048,576 pennies, which is \$10,485.76! Whoa! That's a lot of money, and that's on top of all the pennies you had to give them on days 1 through 19! It's painfully obvious that you should have taken the \$200 option.

## Properties of Exponential Functions

The general form of an exponential function is as follows:

Exponential functions in general form, f(x) = ab x, where b>0, have the following properties:

• The domain of f is all real numbers.
• The y-intercept of f is the point (0, a).
• The graph of f has the x-axis as a horizontal asymptote. That is, the graph approaches the x-axis, but never touches it.
• When a is positive, we have the following: When b > 1, the function is increasing slowly at first and then more quickly, and when 0<b<1, the function is decreasing quickly at first and then more slowly.
• When a is negative, we have the following: When b > 1, the function is decreasing slowly at first and then more quickly, and when 0<b<1, the function is increasing quickly at first and then more slowly.

The last two properties give way to the following general shapes of an exponential function.

We can use these properties to graph exponential functions.

## Graphing Exponential Functions

Let's talk a bit about graphing exponential functions using the properties of exponential functions. Look back at our initial example's exponential function f(x) = 2 x. This function has coefficient 1, base 2, and exponent x. Therefore, it has the following properties.

• Domain = all real numbers
• y-intercept = (0, 1)
• The x-axis is a horizontal asymptote
• Since a = 1 is positive and b = 2 > 1, the graph is increasing and takes on the following general shape.

Putting all this together, we have a general idea of how the graph will look now. We know the left side of the graph will approach the x-axis, the y-intercept is (0, 1), and the right side of the function will increase more and more quickly. Thus, to graph the function, we plot the y-intercept and a few other points that satisfy the function, say (1, 2), (2, 4) and (3, 8), and then connect the dots using a curve that takes on the general shape of the graph.

That's not too hard, is it? Being able to graph exponential functions makes analyzing the function much easier.

## Another Example

An area that exponential functions often show up in is finance, especially in interest problems. A well-known exponential function is the function representing the amount of money in an account after x years, when interest is compounded yearly. This function is as follows.

For example, suppose you were to invest \$200 in an account paying 5% interest compounded yearly. The amount in the account after x years would be represented by the following function

f(x) = 200(1 + .05) x

We can simplify this to get the function in general form and then graph it.

f(x) = 200(1.05) x

In this function a = 200, b = 1.05, and the exponent is x. This gives way to the following properties.

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