 Welcome back to our lecture series Math 1050, College Algebra, for students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misteldine. I want to continue our discussion of power functions in section 4.1 from our lecture series here. Chapter 4 is going to be all about polynomial functions, and so monomials are going to be our first focus in this chapter here. Instead of tackling just monomials, we're going to focus on the general idea of a power function. We say that a function is a power function, basically, if it's a transformation of the form y equals x to the n here, where n could be any real number whatsoever, so we're taking a power of x. This is going to be by power function, so we might slap in a coefficient in front of that a. We could maybe shift it. All those are acceptable here for our power functions. Now, of particular importance in this chapter, number 4 here, we're going to focus on when n is a positive integer, for which we then call this a monomial function. In this lecture alone, we'll focus a lot on monomials, but the general power function will be treated in this situation. Some things I wanted to show you. First, we're going to switch over to Desmos for a little bit to do some graphing. If we consider the function y equals x to the a here, and you were to graph it, you see right here a function where only the domain x equals 0 and greater is included. Now, the reason why I'm going to limit the domain for a little bit is issues like the following. Let's say that we have a power function like g of x equals x to the one-half power, because, again, the x one, it could be any real number we want. The issue is like, what happens if you take g to the negative one, or g of negative one here? This would be negative one to the one-half power, which, as a reminder, when you take fractional powers, this actually gives us a square root. In this case, it would be a square root. It gives you radicals in general. So in this situation, the function outputs the number i. So with quadratic functions in this series, we did see that we do allow for non-real solutions to equations and such. But when it comes to graphing, we are going to stick with the domain convention that we've introduced earlier, that we only will accept real numbers inside the domain. But then we also will only accept those numbers in the domain so that the output is likewise a real number. So for the power function y equals x to the one-half, we would only allow non-negative numbers. Now, that's not going to be the case for everything. Like if we take f of x to be x squared, for example, well, then there's no issue with negatives, right? So you get negative one squared. It's just going to be a positive one right there. No big deal. So what I'm trying to say right now is let's just kind of consider the general power function. And so we can talk about all of them in harmony. Temporarily, I'm going to restrict the domain just to be positive numbers because I want to show you a pattern that happens. If you take a general power function right now, I set the exponent to be one. So this is just y equals x at the moment. One thing that's going to be true about power functions is that this power function, assuming there's no shifting whatsoever. We didn't shift it up. We didn't shift it down. No left or right shifts. Your standard power function, it'll go through the origin zero zero because if you take zero to any power, you are going to get zero. Likewise, this graph will go through the point one one because again, if you take one to any power, you're going to get back one. So those are going to be points that are true for every power function. Now what happens, see what happens as I increase the power. If we allow the power to get bigger and bigger and bigger, you can see that the curve starts bending upward. Right. So we take it all the way up, for example, x to the 10th. You see this type of behavior right here. And so what's happening is the following. When you get past x equals one. Right. So if you get past x equals one, this thing is going to get steep and steeper and steeper. It's steep, steep, steep, steep. And this happens the bigger, the bigger that you make things turn out to be. The bigger the power is, the steeper this part gets. On the other hand, though, when you get close to the origin, it's going to get flat. The bigger the power is when you're between zero and one, it's going to get flatter, flatter, flatter, flatter. But when you get away from one, it's going to get steeper, steeper, steeper. So in general, you're seeing this type of behavior that you have zero over here and one over here. It kind of looks really flat when you're between zero and one, it looks really steep when you get past one. Again, this is what happens when you increase the power to be bigger and bigger and bigger. All right. So this is for the, this is something true for power functions. Now, what happens if you take a small power? If you take like small powers, right, going back to one. At A equals one, it's actually going to be a perfect line. But then if you pick something less than one, it actually bends downward. And we see we actually have the reciprocal behavior going on here. Let's take, for example, A to the point one right here. What happens in this picture is that when you get past one, it gets really, really flat. But when you're close to zero, it gets really steep. So in general, you get something like this. Again, the exact opposite of what we saw a moment ago. This is when you had a big A and this is right here when your A is a small value. That is something between zero and one. And this, these two dimensions will change. This will get more distorted. The bigger, smaller you make the A values get. And another thing to mention here is what happens when you get a number less than one. If you're less than one, you see something happening here. Let me erase this other part that's still on the screen. I'm also going to zoom in a little bit so you can see it a little bit better. When you get a value less than one, kind of what happens at zero is you get a perfectly horizontal line. It doesn't touch the origin anymore. Because if you take something that's zero with power, remember that's going to be like A to the zero. That's going to equal a positive one always. You just get this horizontal line in that situation. And so when you get something that has a negative power, you see this very different behavior. It gets really steep when you are to the left of X equals one. It gets really flat when you're to the right of X equals one. So you're getting something like this. So you get all these different shapes going on here. But what's going on is that as you get closer and closer to X equals zero, you don't actually touch X equals zero in this situation. Because negative powers mean reciprocals. If you have X to the negative A, this really means one over X to the A. And so when you plug in X equals zero, you're going to divide by zero and get... You would actually get something undefined at that moment. This function is going to have something called an asymptote. Something we'll talk a little bit more about in the future. Again, this is just kind of giving you a general idea of how these power functions behave. In this lecture, though, we want to focus on two specific families. So the first one is what happens when we take only odd integers, positive integers. So take, for example, n equals one, n equals three, n equals five. The standard graph Y equals X to the one you can see in front of you right here. Let's label this. So this would be Y equals X to the first. This is just a standard line. It takes the point zero, zero, zero and one, one like before. It also goes to the point negative one, negative one. As we increase the power to something like X cubed, we see that the graph will change. We get something like this. So now there's some curvature that's going on here. We can see that the graph still does go through zero, zero, one, one and negative one, negative one. That part is still the same. But the things, the things, the graph got steeper when you got away from one and when you got away from negative one, but it gets flatter when you're closer to X equals zero. And this is going to happen when we increase the power some more. So if we go up to Y equals X to the fifth, you see this thing happening again. It gets steeper near one. It gets flatter at the origin. Let's do this again. You get even bigger. Here's seven. X to the ninth, X to the 11th, X to the 13th, X to the 15th. Again, this behavior you see happening over and over again. When you get far away from the origin, that is when you get past one or negative one on the other side, the graph gets really steep, but it gets really flat near the origin. Let's watch this again in real time. So we're going to get three, five, seven, nine, 11, 13, 15, 17, 19. I hope I didn't skip anyone there. And so these things will get distorted the more we increase. And this is always for odd powers. Be aware that if you take a number to an odd power, positives are pretty obvious here. But if you take like a negative number here to an odd power, the thing is since you have an odd number of negatives, like if you take three, for example, negative, negative, negative, this is a triple negative. And so the net effect is you're going to have a single negative when you're done. The odd power of negative one is in fact a negative. And so that's why you get some numbers on the left side of the graph there. Let's summarize what we've seen so far with these monomials here. The properties of monomial functions, y equals x to the n when n is a zero. Well, one thing I want to show you is that this graph is symmetric with respect to the origin, right? So we go back to the picture here. Notice that if we were to reflect this graph or if we were to spin this graph around the origin by a half spin, we get the exact same picture again. So this graph is symmetric with respect to the origin. This is will be referred to as an odd function. And actually the word odd function comes from this observation right here. The odd monomials are exactly the odd functions for the monomial families, right? The domain is going to be all real numbers and so is the range. So coming back to the picture right here, you can see that going to the left to the right, it doesn't matter. This is going to be defined for all real numbers. There's no issue with domains. Also, this can go as far up as you want and as far down as you want. We see that as X goes to infinity, Y will go to infinity. And as X goes to negative infinity, we see that Y will go towards negative infinity as well. So the domain is going to be all real numbers here. This graph will contain 0011 and negative 1, negative 1 like we saw. As the exponent N increases in magnitude, the graph becomes more shallow near the origin. It gets flatter and it gets steeper when we get away from the origin. So when you're past 1 or negative 1 like we saw before, we just talked about the in behavior a moment ago. And so this gives us, this right here gives us the standard picture for an odd degree monomial. The basic picture is going to look something like the following. It's flat near the origin, steep and steep. This is what an odd degree monomial looks like. What about even degrees? So let's turn this one off. Let's turn on an even degree. So right now you see the graph Y equals X squared. Notice some important differences here. This function is actually an even function. That is, it is symmetric with respect to the Y axis. That's actually why we call even functions even because even monomials are like the typical even function. It's symmetric with respect to the Y axis. Some other things to notice. It does go through the point 00. It does go through 00. It does go through 11. But this one goes to the point negative 11. It doesn't go through negative one, negative one down here. It actually goes through negative 11. And so notice that as X goes to infinity, the right hand side of this thing is going to go up. So we see that Y will likewise go towards infinity. Now as X goes to negative infinity, notice the graph is still going up. And so we see that Y likewise goes to positive infinity in this situation. We don't see this opposite here. They're both pointing up on this graph. So that's another important difference with this graph. But some things that are the same, notice that as we increase the power. So right now you see Y equals X squared. If you see Y equals X to the fourth, you'll notice the same pattern happens. The points that are far away from the origin got steep. And the stuff that's close to the origin is going to get flat. Let's look at the next power, Y equals X to the sixth. You'll see it again. The part near the origin got flatter and the parts away from the origin got steeper. And this will just continue to happen. X to the eighth, X to the tenth, X to the twelfth, X to the fourteenth, sixteenth, eighteenth, twentieth. That same basic pattern is happening. You get steep here and you get flat here, making sort of like this bucket shape when we talk about an even monomial function. So let's summarize what we discovered right here. The properties of a monomial function when Y equals X to the n and n is an even number. Well, in terms of symmetry, it'll be an even function. This is actually why we call even functions even because they're mimicking this behavior right here. The domain is going to be all real numbers, but it turns out the range is not all real numbers. The range is only going to be non-negatives. Coming back to the picture right here, you'll notice that in terms of our domain or range, the domains for all real numbers, you can take any power of a number and that'll be a number again. But you don't get anything below the X axis. Because of issues with complex numbers and things like that, if you square a positive, or if you take a positive to an even power, you're going to get a positive. Zero to any power is zero. A negative raised to an even power gives you something positive. And so it's not possible with a real input to produce a negative output for these even monomials. So its range, in fact, its range is only going to be zero to infinity. We'll contain the point zero, one, one. We also get negative one, one in this situation. This part's also true for all power functions that as the end gets bigger as you increase the exponent, it'll get shallow near the origin, but it gets steep away from the origin. And as X goes to infinity, like we saw, Y goes to infinity. And as X goes to negative infinity, your graph also will still approach infinity right there. And so that describes some basic properties about the graphs of monomial functions. In the next video, I'm going to take a look at how we can apply graph transformations to these basic pictures and start graphing some very primitive polynomial functions.