# Comparing Linear & Exponential Functions

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• 0:04 Making Money
• 1:22 Constant Change
• 2:23 Percent Change
• 3:43 Exponential Versus Linear
• 5:06 Lesson Summary
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Lesson Transcript
Instructor: Stephanie Matalone

Stephanie taught high school science and math and has a Master's Degree in Secondary Education.

In this lesson, we will go over the definition of linear and exponential functions then compare and contrast the two. We will talk about how to decide if a function is linear or exponential and go over examples of each.

## Making Money

You parents tell you they will give you ten dollars a day for the rest of the year or give you one penny and double it for the rest of the year. You get to decide! Hmmm...a penny sounds like nothing, so you decide to go with the first option. Uh oh...big mistake!

Getting ten dollars a day for the rest of the year is an example of a linear function. Linear functions are functions in which the rate of change is constant. Because you are getting the same amount of money each day, your change in money is the same.

Doubling your initial one cent every day is an example of an exponential function. Exponential functions are functions in which the the rate of change is not constant (not adding the same ten dollars each day, in other words). In this case, the rate of change increases each time because you are getting more money each day (doubling your money). Exponential functions increase by the same percent each time. In your case, your money increases by 100% each day.

Why did you make a mistake? Because the money is doubling, it will increase very fast. It might seem slow at first. After day one, you only have 2 cents; after day five, you only have 32 cents; but after day 30, you already have \$10,737,418.24! Compare that to getting ten dollars a day and by day 30, you only have \$300.

## Constant Change

So what is this rate of change business? When talking about functions or graphs, we think of rate of change as the movement in y versus the movement in x. When we say a function has a constant change, we are saying that when we move one unit in the x direction, we will always move the same distance in the y direction. In other words, if we move one unit to the right and two units up, we need to keep moving two units up every time we move one unit to the right.

You might know of this as the slope of a line or linear function. Slope is simply the change in y divided by the change in x. For a function to have a straight line, it must have a constant rate of change and have a slope. This makes it a linear function.

You can see a constant rate of change with the function y = 3x + 1. Every time x goes up by one, y goes up by three. The same amount is added to y each time x changes by one. The slope will be equal to the change in y over change in x, which is 3 / 1, or 3. This means the linear function will be graphed as a straight line.

## Percent Change

When you look at the function y = 3 x , you can see that the y values do not go up by a constant number being added, therefore it's not a linear function. It is an exponential function because the y values are increasing exponentially. Instead of adding the same number to the y values, we are increasing the y values by the same percent each time. In other words, we are multiplying the y values by the same number.

With exponential functions, we do not use slope, but rather percent change, or how a variable increases or decreases. We use percent change with exponential functions because the y values are increasing or decreasing by a certain percentage for each change in x. It sounds like the constant change from before, but remember that constant changes must come from ADDING the same number to y.

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