Exploring changes over equal intervals

0:00- [Instructor] So I have the graphs of two different functions here. I have the function g, and I have the function h here. And you immediately see they look very different. Some of you might immediately be thinking, "Hey, this one looks linear. This looks like a straight line.

0:13This one looks, well, definitely non-linear." Maybe you already know this is exponential. But what we're gonna do in this video is interpret a little bit more about some of the properties or ways that you can tell linear and exponential functions. So for example, on this first graph right here, this first function when x increases by 2., so maybe if I say x goes up by 2, what does that do to the value of the function?

0:42Pause that. Pause the video and think it through. All right, so if I increase x by 2, so if I go from say x equals 0 to x equals 2, it looks like the value of the function has increased by, went from negative 2 to 4, it went up by 6. If I increase by 2 again, it's going to go up by 6 again, and this is true of a linear function.

1:13If you have an equal interval that you're increasing by, you're going to have the same amount of difference between the corresponding values of the function. And you probably already recognize this when studying slope. In order for the slope at every point of this line to be the same, this actually has to be true.

1:30So what you could say, when x goes up by 2, g goes up by 6. Maybe I'll say g goes up by 6. I'm kind of doing my own little notation here. But no matter where you start, as long as you increase x by 2, g is going to go by 6. Now, you could have also said, "Hey, if I increase x by 1, what happens?"

1:51Well, if I increase x by 1, it looks like g goes up by 3. If x goes up by 1, g goes up by 3, x goes up by 1, g goes up by 3. So one of the properties of a line of a linear function is that if you increase your input in this case x by a fixed amount every time, the same interval, or you're going to change your function by, not necessarily that same amount, but no matter where you start, by a certain constant amount.

2:23Now we can move over here and we can think about something analogous. Let's think about what happens when we increase by 2 over here. So for example, if I go from x equals 0 to x equals 2, what happened to the value of the function? The function went from 1 to 4.

2:48So you might say, okay, it went up by 3, if you're trying to do the same type of analysis, but if you increase by 2 again, it hasn't gone up by 3 now, it's now gone up by a much larger amount. It's gone from 4 to 16. It's now gone up by 12. So in a non-linear function, you're not going to necessarily have that property that you're always for a given fixed interval and x, that you're going to have a steady change in your y or in the value of the function.

3:17And the way that you recognize an exponential is, instead of looking at how much you would increase your value of your function, you should think about what is the factor, so what are you multiplying every time? So lemme get rid of this plus 3 and plus 12. But let's think about it in terms of multiplication.

3:33To go from 1 to 4, as we increased our x by 2, you could multiply by 4. Similarly, when you increase x by 2 again, we're going from the value of the function being 4 to the value of the function being 16. Once again, you multiply by 4. So in exponential functions, for a given fixed intervals that you're changing your x over, you're going to have a constant factor.

3:59You're gonna be multiplying by the same amount. While with a linear function, by a given fixed shift or fixed change in x or fixed intervals in x, you're going to have a constant that you're adding to successive values of your function.