Examples of linear and exponential relationships

So I have two different XY relationships being described here, and what I would like to do in this video is figure out whether each of these relationships, whether they are either linear relationships, exponential relationships, or neither. And like always, pause this video and see if you can figure it out yourself.

So let's look at this first relationship right over here. The key way to tell whether we're dealing with a linear, exponential, or neither relationship is to think about, okay, for a given change in x. And here, you see each time here we are increasing x by the same amount. So we're increasing x by three.

Given that we are increasing x by a constant amount, by three each time, does y increase by a constant amount? In which case, we would be dealing with a linear relationship. Or is there a constant ratio between successive terms when you increase x by a constant amount? In which case, we would be dealing with an exponential relationship.

So let's see here. We're going from negative two to five, so we are adding seven. When x increases by three, y increases by seven. When x is increasing by three, y increases by seven again. When x increases by three, y increases by seven again. So here, it is clearly a linear relationship.

In fact, you could even plot this on a line. If you assume that these are samples on a line, you could think even about the slope of that line. For a given change in x, the change in y is always constant. When our change in x is 3, our change in y is always 7. So this is clearly a linear relationship.

Now let's look at this one. Let's see, looks like our x's are changing by 1 each time, so plus 1. Now, what are y's changing by? Here, it changes by 2, then it changes by 6. All right, it's clearly not linear. Then it changes by 18. Clearly not a linear relationship.

If this was linear, this would be the same amount, same delta, same change in y for every time because we have the same change in x. So let's test to see if it's exponential. If it's exponential, for each of these constant changes in x, when we increase x by 1 every time, our ratio of successive y should be the same. Or another way to think about it is, what are we multiplying y by?

So to go from 1 to 3, you multiply by 3. To go from 3 to 9, you multiply by 3. To go from 9 to 27, you multiply by 3. So in a situation where every time you increase x by a fixed amount—in this case, 1—and the corresponding y's get multiplied by some fixed amount, then you are dealing with an exponential relationship. Exponential! Exponential relationship right over here.