 In this video, I'm going to talk about recognizing linear functions. So basically what we're going to do is we're going to look at some data, we're going to look at just some numbers, no pictures or anything like that, just numbers, and try to figure out if this is a linear function. So is this a function that goes basically in a straight line? It's always increasing, always decreasing at the same rate, something to that effect. Okay, so a couple of examples here. Determine whether each data set, again, we're dealing with data here, not numbers. Determine whether each data set could represent a linear function. Okay, so what we want to do is we want to look at the X's and the Y's, or the X's and then the function itself. So these X's are our inputs, and then this function is the output, input output. So these are the numbers we put in, and these are the numbers that we get out. So if I plug in a negative 2, I get a 2. So again, you don't see an equation here, we just see numbers, we just see data. We don't know what the equation is. If we knew what the actual function was, if we knew what the equation was, we could just look at it and figure out if it was linear or not. But in this case, we have to look at the data first. Okay, so here we go. So I have, what we want to concentrate on is we want to concentrate on looking at the changes in the numbers, or how much they go up and down each time. So not necessarily looking at the numbers, so from negative 2 to 2, if I plug in a 0, I get 1, or plug in a 2, I get 0. I don't really want to concentrate on that aspect of it. I want to concentrate between these two. So if I go from a negative 2 to a 0, I want to kind of concentrate on going between those two numbers. So I'm just looking at my inputs, I'm just looking at my X's, and I'm going to go from one to the other and see what kind of increases I'm getting. So in this case, in the top row, negative 2 to 0, that's going to be adding 2, okay? So from 0 to, 0 to 2, that's another plus 2, and from 2 to 4, that's also another plus 2. Okay, so we can see each time my inputs are just continually going up by 2, up by 2 each time. And that's a good thing. When we're looking for a linear function, okay, when we're looking for a linear, a line function, when you increase, you want to increase at the same rate. Lines increase at the same rate. This actually delves into a little bit about slope, but I won't get into that now. But everything's going to increase or decrease at the same rate. So notice that all of my X's are increasing at the same rate. That's a good thing. Okay, so let's go down and look at my Y's, look at my function, look at my outputs. So I've got a couple different names for it. I'm just going to call them outputs. Okay, so I'm looking at all my outputs, I'm going to go from one to the next to try to figure out how much they're increasing by. So in this case, I'm actually decreasing by one. Okay, now that's okay, decreasing is okay. I just want to figure out what's happening each time. Okay, negative one, and then one to zero is also a negative one. And from zero to negative one, that's also a negative one. So every single time, I'm decreasing by one. Okay, now again, that's a good thing. That tells us that we're on track for a linear function. Okay, so now I look at the top here, I look at the bottom here. The top, it's a constant increase. Everything's increasing by two. Down here, it's a constant decrease. Everything's decreasing by one. So actually, this right here is a linear function. Again, a linear function, kind of the informal definition, is that the data, the numbers, whatever it is, your inputs and outputs either increase at a constant rate or they decrease at a constant rate. So here we see increasing at a constant rate, decreasing at a constant rate. And that rate is increasing by two every time, decreasing by one every time. Everything is constant. So that right there is how you know that that data set is a linear function. Okay, so let's go to the next one, since we have an idea of what a linear function is, what the numbers look like. Okay, so again, I'm going to go look at my inputs, look at my outputs. So input, I'm just going to go through this quickly. From two to three, I'm adding one. Three to four, adding one, four to five, adding one. Okay, went through it pretty quickly, you can identify that pretty easily. And on the other hand, two to four, that's adding two, four to eight, you're adding four, eight to 16, you're adding eight. Okay, now this is, you might start recognizing this a little bit of trouble. On the other example, constant increase. Everything's the same. Down here, constant decrease. Everything is decreasing the same. But everything stays constant. It's the same number. Okay, but over here, okay, we have a constant increase. Okay, so it looks like we're on the right track. But on the other hand, when we go down here, we're adding two, then we're adding four, then we're adding eight. This right here is a dead giveaway. This should set off warning signals. Okay, this right here is not a linear function. Not linear function. Okay, and the dead giveaway, let me change my color here just to emphasize this a little bit, this right here is that dead giveaway. Everything, you're not increasing by the same rate. You're, the increasing, how far up you're going each time, that changes every time. So there's a lot of changes going on here. That's what we don't like. That's what we don't need for a linear function. Okay, so that right there, that blue, that tells us that this is not a linear function. Okay, that's it, that is recognizing linear functions. Hopefully these two examples give you an idea of recognizing just a linear function just with the data. Again, we have no pictures, no graphs, no nothing like that. We just have the numbers. We just have the data. Okay, I'm gonna go on just a little bit more saying about what kind of functions these are. Just going a little bit more depth into what these are. I just wanted to do a couple of examples real quick and then explain these a little bit more. So over here on my first example, notice that everything is increasing. So all of my inputs are increasing and then my outputs are decreasing. So as one of my variables increase, this other output variable is actually decreasing. This is what we call an inverse relationship. And as we go through your math courses, you'll learn about a direct variation and indirect variation. This is what we would call an indirect, write that out, indirect, variation, indirect variation. This basically means as one of my variables goes up, so as my inputs increase, these numbers go up, the other, the outputs actually go down. The graph of such, let me change my colors here a little bit to emphasize this a little better. If I make a very simple graph of this, in the first quadrant here, this would be, so here's my x and here's my, sorry, there's my x axis and this is my y axis or my f of x axis, same difference here. So as my axis, as I increase along this axis, that means my f of x axis is gonna go down. So the graph of this would look something like this. Okay, and those of you who have seen this before, you'll notice that that's a negative slope. So as one of my variables increases, this other variable is going to decrease. So we got a downward slope. Now some real life examples, if I could think of one real quick. A real life example, as I increase something, something decreases, I guess one way to put into it is my level of activity, my level of physical activity and my body weight. Now I know there's a whole lot of other variables in there, but this is just a very basic example. So one way you can think of it as a graph is that as I increase my level of activity, as this, as my level of activity, we'll call that x, as that gets bigger and bigger and bigger, my weight is actually gonna go down. So the more active you are, the lower your body weight is going to be. I just realized I made a mistake here. Hopefully I didn't lose you. Sure, check out that, anybody. Anyway, variation, sorry about that. Anyway, so again, back to my example, as my activities increases, the more active I am, the more hours of activity that I have, the lower my body weight will be. Now I know there's a lot of different variation out there, genetics and diets and all sorts of stuff like that, but you guys get the idea. So that's what we call an indirect variation. On the other hand, this other example over here, we identified that was not a linear function. So if it was not a linear function, then what is it? Well, in this case, and you'll learn this in a different video in a different section, this increase right here, this is actually an exponential increase. And some of you may have already recognized that. Two to four to eight to 16, this is two, this is two to the first power, this is two to the second power, this is two to the third power, and this is two to the fourth power. And if you do the math real quick, two squared is four, two to the third is eight, two to the fourth is 16. This is what we call an exponential growth type of problem, which again will be saved for a later video later, a later section. But anyway, I wanted to go into a little bit more depth of what some of those actually are and what you could recognize them as. But in general, again, just to summarize everything once more. So to identify if something is a linear function, it has to have constant change. It has to be the same change. So we're changing by twos here, very good, and then we're changing by negative ones here. Everything is constant, that makes it a linear function. Over here, we had constant change for our inputs, for our x's, but then once we got down here to our outputs, we noticed that it wasn't constant change anymore, the change itself kept going up and up. So that was how we identified that as not a linear function. So a little summary there. Hopefully this video helped you to recognize linear functions. Thank you for watching.