In this lesson, we will learn about the rate of change of a function with specific focus on linear and exponential functions while examining the graphs of both types of functions.

Determine the Rate of Change of a Function

Do you like doing your chores? How about doing the dishes? I really hate to do the dishes! Here’s an idea to make doing the dishes easier. Tell your parents that tonight you will do the dishes for a penny. I know what you’re thinking. You’ve got to hang in there with me. Just tell them that they have to double your pay every night.

Okay, you still with me? Okay, as long as you’re with me. So, tomorrow night you’ll make the fortune total of .02 cents. Hang in there. You can’t expect to get rich over night; it might take a week or two. Where was I? Oh yeah, day three. On day three you would make .04 cents, day four you would make .08 cents, day five you make .16 cents, day six you would make .32 cents, and on day seven you would make .64 cents! Yay!

.64 cents isn’t bad pay for doing the dishes, but it gets better the second week. On the first day of the second week you’ll get paid $1.28. I think you’re going to like this week. By the end of that week you’ll be making almost $82.00 for doing the dishes! Can you imagine what the third week will be like?

You probably never realized that being a mathematical genius could be so profitable! What we’ve discovered here is the exponential rate of change.

The Rate of Change of a Function

There are two types of rates of change of a function. An exponential rate of change increases or decreases more and more quickly, while the linear rate of change increases or decreases very steadily. An exponential rate of change is represented by an equation involving an exponent. A linear rate of change is represented by an equation that only involves multiplication or addition. An exponential rate of change has a graph that is curved. A linear rate of change has a graph that is, well, linear! Let’s focus specifically on linear rates of change.

Linear Rate of Change of a Function

Many functions are not quite as exciting as doing the dishes! I’ll bet you never thought that doing the dishes could be exciting. Anyways, where was I? Oh yes, linear rates of change. A linear rate of change would be something like the deal most kids have with their parents.

Suppose that your parents agreed that it was your job to do the dishes, and you would get $1.00 each time you did the dishes. You wouldn’t get more the second time. You would just get another. How about the third time, how much would you get paid? You guessed it, you would get paid $1.00. Wait a minute, there’s no such thing as a three dollar bill! That’s better.

Now you’re not going to get rich doing this, are you? But, if you save your money, it still adds up! That’s a linear rate of change. Let’s look at some mathematical functions involving linear rates of change.

Suppose your little business eventually becomes a thriving dish washing business! And you went around the neighborhood washing everybody’s dishes! This is going to be great!

You develop a plan where you charge $2.00 per day plus .25 cents for each dish. That can be written in the equation: y = .25x + 2, where y represents your total pay, and x is the number of dishes.

No matter how many dishes you do, you’re going to get $2.00. But, if you wash 10 dishes, you would have .25 * 10, or another $2.50 to add to your $2.00. As a matter of fact, your profits could be expressed in a table like this.

A linear rate of change has a common difference between each output
Table with a linear rate of change

Do you notice how the y-column of the table goes up by .25 cents each time? This is what we call the common difference. The common difference in a linear function is always constant; it never changes. The common difference is also known as the rate of change.

Do you see the same rate of change in the equation? That means that the rate of change is the number multiplied by the x, or as we know from looking at functions, the rate of change is the number multiplied by the domain value, and this rate of change tells us how much the output or range will go up with each step. This also means that in the graph, for every one space the line moves over, it goes up .25 spaces. Because of this, we say the rate of change is .25.

Lesson Review

A rate of change is how fast a function is changing. These rates of change can be exponential or linear. Exponential rates of change get faster and faster as they go along, while linear rates of change proceed at a constant rate. Exponential rates of change come about from multiplying repeatedly, and linear rates of change come about from repeatedly adding the same thing.

Exponential rates of change have a graph that is a curve. Linear rates of change have a constant rate of change that results in a straight line. You can determine a rate of change from a graph by looking at how much a function increases or decreases over one unit. Well, I guess that you need to get started on those dishes! Personally, I use paper plates!

Learning Outcomes

After this lesson is done, you should be able to:

  • Name and describe the two rates of change for functions
  • Identify a graph as having a linear or exponential rate of change