 Welcome back to our lecture series Math 1050, College of Oxford for Students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misseldine. As you can see, this is our first video for lecture two in our series for which we're continuing on with our discussion of function fundamentals. Last time we had talked about four ways of representing functions. We can represent functions graphically, numerically, verbally, or algebraically. Like we mentioned in the previous video, we were avoiding both the verbal and algebraic representations of functions for now. Algebraic representation, although it seems quite natural for a college algebra class, we'll get to those eventually. But I want to delay just for a little bit longer, so we can focus more on the nature of the functions themselves and not get distracted by the algebraic formulas associated to them. In these videos and lecture two, we're going to do some more examples with tabular functions and graphic functions and try to describe these properties and definitions associated to functions in terms of more visual representations. Let f be a function defined on some interval, and let's say that the open interval, a comma b, is a subset of the domain of our function f. With those assumptions in mind, we say that f is increasing on this interval a to b. If whenever x1 is less than x2, that implies that f of x1 is less than f of x2. Recall, f of x1 is the y-coordinate connected by the function f to the x-coordinate x1. Likewise, f of x2 is the y-coordinate connected to x2 via the function f. Similarly, we say that the function f is decreasing when whenever x1 is less than x2, that implies that f of x1 is greater than f of x2. The difference here between increase and decreasing that I want to point out is that as you move left to right, so saying that x1 is less than x2 on the x-axis, that means you're reading it to the right, just like how we read books in the English language, we read left to right. As you read the graph left to right, the x-coordinates move from left to right, this says that the y-coordinates are getting bigger. As the graph moves from left to right, the y-coordinates are getting bigger, bigger, they're going up along the y-axis. Decreasing does the opposite. As you read the graph left to right, the function's actually getting smaller. The graph is going down along the way. Similarly, we say a function is constant whenever f of x1 equals f of x2, even when the x-coordinates differ with each other. In a nutshell, as we read the graph left to right, if the graph's going up, we say it's increasing. If the graph's going down, we say it's decreasing. If the y-coordinates are changing at all, we say that it is constant. If the function's increasing or decreasing, we say that it's monotonic on the interval. So in the English language, the word monotone typically means sort of like the following example. If someone's talking monotone, that means their voice level never changes, though with the idea of increasing, decreasing, a function we say is monotone, if it's always going up or it's always going down. That is monotone means there's no change in this regard. So if it's always going up or always going down, we say that the function is monotone or it's monotonic. If there's no change in up and down with the behavior of the graph, that of course means it's constant. Now, a similar definition associated to monotonicity is the idea of concavity. We say that a function f is concave upward on the interval a to b. If the graph of f curves in an upward direction, so you might see something like the following. A graph would be concave up on an interval. If you see this type of upward curvature, it bends up like this. Similarly, we say that f is concave down, downward on an interval a to b. If the graph curves in a downward direction, you get this downward bowl kind of shape like this. This would be concave down. And I keep on saying bowl. And the idea is, if you kind of imagine this graph is some type of bowl, would it hold water or yummy soup or whatever you wanna put inside of it? If it would hold the water, then we call that concave up. On the other hand, if you pour your bucket on someone's head, maybe to raise awareness of a disease or something like that. If the water would to fall out of the bucket, then we would say that function is concave down on that interval. It turns out there is a third option. The function might not curve upward. It might not curve downward. It actually could be straight. There might be no curvature to the graph whatsoever. And so concavity is measuring this curvature. Is it bending up? Is it bending down? Or is there no curve to it at all? It could be straight, like a line. Monotonicity is about whether the graph is rising or falling or maybe it stays flat line because it's constant. So let's look at an example of such a thing. Consider the following graph right here. If we want to determine its behavior about monotonicity and concavity, we might follow the following train of thought. Let's do the monotonicity first. We wanna identify where is the function increasing? Where is the function decreasing? And if anywhere, where is the function constant? Note that the answer to the question is like this. If you asked where the function's increasing, where is it decreasing? These are going to be written, first of all, as intervals. So we'll put them as open intervals. If we look at the left side of the graph starting here at negative six, notice what happens to the graph as we go from negative six down here to negative four. The Y coordinates are falling as we move from left to right. So this indicates to us our function is decreasing. It's decreasing on the interval negative six to negative four. You'll notice, of course, that when I wrote this interval negative six to negative four, I put parentheses on it as opposed to brackets. And that's because we're neither including the number six, negative six, or we're also not including negative four inside of the interval. When you look at negative six, for example, could it really be decreasing at negative six? You could ask that question, right? I mean, if there's nothing to the left of negative six, how do we know if it's getting smaller? Then also, when you look at negative four, it's kind of like, well, it was decreasing to the left of negative four, but we're gonna see in a second that it's increasing when you get past negative four. There's something special about negative four, and it seems to be sitting on the fence. At negative four, it's neither increasing nor it's decreasing. So these places where it switches, monotonicity, we don't include those in the interval. Like I mentioned a moment ago, once you get past negative four, it's gonna start to increase, right? So this function's gonna get bigger, bigger, bigger, bigger, bigger until you hit zero. And so we can actually notice here that the function is increasing from negative four to zero. In all of these cases, we're recording the X coordinates, not the Y coordinates. We need to know the address of these intervals on which part of the domain is it increasing, decreasing. When we hit zero, X equals zero, you'll notice that the function flatlines. This is actually a constant section of the function, so that the function is actually constant on the interval zero to three. And then at three, it switches again, and from three to six, we see that it's decreasing again. So there's actually two sectors on the graph where the function was decreasing. They're disjoint intervals, and so we're gonna use the union symbol to connect them together. We say this function was decreasing from negative six to four, union three to six, like we see right there. Now let's consider the problem of concavity. It's kind of similar to monotonicity, but it's not the same thing. Where is this function bold shape up? Where is it bold shape down? Or where is it possibly straight? So you'll notice here, starting at negative six, I've read the graph left to right. As we go from negative six to negative four, we can actually go past negative four. We actually have our bold shape rover right here, which is not blue this time. So it can't be water. It's some type of icky witch's brew or something. I don't know, but the bull is holding the witch's brew right here. It's a bold shape up. This is concave up sector. So we would say the following, our graph is concave up on the interval negative six to negative two. It does appear somewhere around negative two. It switches the curvature. From approaching negative two, we see that the curvature is curving up. But when you get past negative two, the function seems to be bending downward. You'll notice that it's increasing to the left and to the right of negative two, but it's the curvature that changes. From negative two to zero, the function seems to be bending down. If we kind of continue in this trajectory, we might get this type of bull shape. You're pouring water on someone's head, or in this case, which is brew on someone's head. And so we would actually record this here that we are concave down from negative two to zero. When you go from zero to three, that one's straight. Whenever you're constant, you necessarily have to be straight. But of course, if the function is straight, that does not necessarily mean it's constant. For example, if you just take any line, you could have an increasing line, you could have a decreasing line. Those are straight, but they're not constant. And then the last sector, we have a concave upward shape again. Notice how the curvature is going up, even though the function's decreasing. So we actually, we record this here, union. We were concave up from negative six to negative two, and then we're gonna union that with three comma six, like so. And so we're able to identify where the function was increasing and decreasing, where it was at constant. That's its monotonicity. We're also able to determine where the function was concave up, concave down, and also where it was straight. Now it turns out the two ideas of concavity and monotonicity are distinct from each other. So it's important to pay attention to these on the graph. And you can see these different behaviors are happening on this graph. Was there anywhere on the graph that it was increasing and concave up? If you were increasing concave up, you would look something like this. Notice this function here is increasing in concave up. Yeah, sure, that happens right here from negative four to negative two. It was increasing and it was concave up. Was there anywhere where it was decreasing and still concave up? Decreasing concave up would look something like the following. And sure enough, we actually see two places on the graph from negative six to negative four was decreasing concave up. We also see from three to six, it was decreasing concave up. Was there anywhere on the graph where it was increasing concave down? If you were increasing concave down, you get a picture that looks something like this, which is demonstrating on the graph from negative two to four. Another possibility is, could the graph be decreasing concave down? If you're decreasing concave down, you get a picture that looks something like this, which actually this graph doesn't demonstrate that anywhere. There's no decreasing concave down intervals. The only other thing we haven't considered yet would be that it's straight in this sector right here. So this graph is a pretty good example of demonstrating most of the possibilities you can get here. But what I want you to get from this picture is that these issues of concavity and monotonicity are independent of each other, related, but they're not the same thing. One does not imply the other. And so this is a pretty good example to investigate these definitions.