 Welcome back to our lecture series Math 1050, College Algebra for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In section 5.2 in our series, we're going to be talking about rational graphs. That is, we want to graph rational functions, which we introduced in the previous lecture here. Now, in order to do that, we have to have a baseline. When it came to graphing polynomials, we compared polynomials to monomial graphs. So we started off by graphing things like y equals x squared, y equals x cubed, y equals x to the fourth. We want to do the same basic thing for rational functions. Now, our baseline is going to be the reciprocals of these monomial functions. We want things like y equals 1 over x, y equals 1 over x squared, y equals 1 over x cubed, 1 over x to the fourth, 1 over x to the fifth, 1 over x to the sixth, keep on going. So it turns out just like monomials that there's going to be a parity argument, that those with an even power are going to behave very differently than those with an odd power. So I want to illustrate this by flipping over to Desmos for a second. So imagine we have the following graph in front of you right here. This right here is the function y equals 1 over x. And so this is the standard reciprocal function. Some things to notice in this graph right here is that the point 11 lives on the graph. The point negative one, negative one, likewise lives on this graph. These were points of importance when we talked about odd monomials previously. Now, something that is different here is that the point 00, the origin, is not a point on the graph. It actually lives outside of the function's domain. Because if you try to plug in x equals zero, you get 1 over zero, which is undefined. And so our function, our rational function here is not defined at x equals zero. So it's definitely not the origin, but it's nothing. There's nothing there. And so something we see on rational functions is you're gonna see the presence of vertical asymptotes. So in fact, there is this vertical line, which in this case coincides with the y-axis, that our function seems to avoid. That as we closer and closer to x equals zero, either on the right or on the left, your function is going off towards infinity, like on the case from the right, or you also get the y approaches negative infinity in the case of the left. And so our function, it's not defined at x equals zero, and we see there's this asymptotic behavior. This is what we meant in our previous lecture about vertical asymptotes. We talked about how we can algebraically find those vertical asymptotes. And so now we can see that in fact, there is this geometric interpretation. We see what happens as we approach this point outside the domain. This is something we'll investigate more for more complicated graphs. But our function also has a so-called horizontal asymptote that in this case it coincides with the y-axis. That as x approaches infinity, we're gonna see that y approaches zero. And as x approaches negative infinity, that as you go to the left side of the graph, y likewise approaches zero. Now to try to distinguish between these two pictures, you'll notice that on the right-hand side that as we get closer to the x-axis, it's from above. And so we're gonna denote this as zero plus, the little superscript of plus means that we're approaching zero from above. We're approaching it from the positive side. But then on the left-hand side as we approach zero, in this case, we see that as we approach zero, we're approaching it from below. That is, we're approaching it from the negative side. And so we're gonna denote this as zero negative in the superscript. So these are not exponents, just a little bit of notation. We're borrowing from calculus to indicate that we're approaching zero from above on the right and we're approaching zero from below on the left. And it turns out that for the most part, this behavior I've described to you can describe any odd monomial function. So as we, right now you see one over x to the first on the screen, as we then switch this to be one over x cubed, right? So that's what we're looking at right now, one over x cubed here is our y-coordinate. Notice what happened as we went back and forth that some things stayed the same, right? This graph still passes through one, one and negative one, negative one. So those two points were the same. The horizontal and vertical asymptotes were the same. And notice where did the vertical and horizontal asymptotes intersect? They intersect at the origin, right? So that's the significance of the origin here. It's not a point on the graph. This is where the asymptotes intersect each other. So as you go between one over x and one over x cubed, that behavior stays the same. As you go off towards infinity, you're still approaching zero from above on the right and you're still approaching zero from below on the left. That part's the same. So the in behavior is the same. And as you approach your vertical asymptote from the right-hand side, you go off towards positive infinity. As you go off towards, as you approach the vertical asymptote from the left, you're gonna get negative infinity there. And so again, a bit of notation here. Let me mention that we would say that as x approaches zero from the right, y approaches positive infinity. In this case, as x approaches zero from the left, y will approach negative infinity. So the behavior near the asymptote is the same in both of these situations. Basically the only thing that changed as we went from one over x to one over x cubed here is the steepness. That when you are close to the vertical asymptote, it gets steeper. And when you get farther away, you're actually getting closer to the horizontal asymptote faster. That is, it's more shallow when you get past one and negative one, but when you're between one and negative one, it gets steeper. And this will happen as we increase the power. So we have one over x to the fifth, one over x to the seventh, one over x to the ninth, and to the 11th, all the way up to the 19th power, right? What happens is when you get close to the origin, it gets steeper right here. And when you get far away from the origin, it gets even more shallow right there. And so that's what happens when you have these odd monomial functions. And so I want to summarize that for a second. So some things we should mention that if you have an odd, not monomial, an odd reciprocal function, y equals one over x to the n, where n is an odd number, like one, three, seven, et cetera. Some things to note here is that the graph is symmetric with respect to the origin. This is what we called an odd function. Let's switch back over for a second, right? Notice that this function is in fact odd, that if we rotate around the origin by a half spin, we get back the exact same picture. Or another idea, if we take the line through the origin, we find a point equidistant on the other side. So this is an example of an odd function. It's symmetric with respect to the origin. The domain is gonna be all real numbers except for zero. We saw that there actually was a vertical asymptote at that point. The range is also all real numbers except for zero. We can get any number coming out of this reciprocal function except for zero itself. We mentioned there was a vertical asymptote at x equals zero, horizontal asymptote at y equals zero. It goes to the points one, one, and negative one, negative one. As the exponent increases, it got steeper near the origin and it got flatter when you're away from the origin. We mentioned that. And then we also mentioned the behavior near the asymptote that you have your vertical asymptote. It'll approach from the right-hand side as x approaches zero from above, y will approach infinity. And as it'll go down like this, as x approaches zero from the left, y will approach negative infinity. So we can determine the asymptotic behavior near the vertical asymptote. That's the same for all of these odd reciprocal functions. And then as, if you think of your horizontal asymptote, as you go to the far right as x approaches infinity, y will approach zero from above. And as you go to the far left, as x approaches negative infinity, y will approach zero from below. And so with the in behavior and the asymptotic behavior of our functions, determined just by this parity, just by it being odd. Now let's talk about even ones for a second. So let's switch this thing up. We're gonna go from two to say 20 this time. And I'll start off at two. So some important things to see here that the behavior of the graph did change a lot. So first of all, the point one one is still on the graph, but negative one, negative one's not on the graph, which isn't so surprising because for even monomials, negative one, negative one was never a point on the graph. Instead, the point that we won't want would actually be negative one, positive one, which you can see right there. That was a point that was typical for our even monomials, like if you take y equals x squared for a moment, you see this parabola here on the screen. Those points go in the same, they go in the same on both graphs right there. You see both of them present. Now, one thing that's interesting to mention here, I'm gonna switch the color to make it a little bit more clear on the screen. Now, the parabola is the orange one, that these functions are reciprocal to each other. So as one number gets big, the other ones have to get small. And so you see what happens that when the parabola gets small, that's getting close to zero, then the reciprocal function has to get big, going off towards infinity. Now, as the parabola gets big, going off towards infinity, its reciprocal has to get small because that's what reciprocation's all about. When one gets big, the other gets small. All right? So coming back to the graph right here, the even reciprocals will go through one, one and negative one, negative one. They don't go through zero, zero, the origin, but they do have a vertical asymptote at x equals zero and a horizontal asymptote at x equals zero. As the power gets bigger, right? You're gonna see the same thing about steepness, right? As we go up bigger, bigger, bigger, bigger, bigger, bigger, right? It's getting steeper when you're close to the origin and you get flatter when you're away from the origin, the exact opposite of what our monomial functions did. Okay, we're gonna go back to one over x squared here. Another thing important to remember, let's look at this asymptotic behavior. So as you get close to the origin, as you get close to the vertical asymptote, the graph right here is x approaches zero from the right. You're gonna see that y approaches positive infinity. That was the exact same truth statement for the odd reciprocals. But in this situation, notice that as x approaches zero from the left, the function still approaches positive infinity and this is something we see all the time that with the odd powers, you'll see sign switching. You had positive versus negative, but with even powers, you don't see the sign switch, right? Negative one squared is actually a positive one, but like negative one cubed is in fact a negative one. Odd powers will keep the negatives, even powers will make them positive and we see that same feature happening right here. As x goes off towards infinity, as x goes to infinity, we see that y will approach zero from above, it's positive, but as on the left-hand side, as x approaches negative infinity, let me fix that, as x approaches negative infinity here, notice that y will still approach zero from above. It approaches it from the positive side. So when we take even powers of negative numbers, it makes them positive and in fact, this is an even function is symmetric with respect to the y-axis there. All right, so let's go back to our picture. Oh, sorry, let's summarize what we've discovered here about these even reciprocal functions. So they are even functions, so they are symmetric with respect to the y-axis. Their domain is gonna be all real numbers except for zero. I do need to make a little, that's not correct what it says about the range. The range in this situation is actually gonna be zero to infinity. It's only gonna be non-negatives. And I also have to make one more amendment there. Sorry about that. It should be zero to infinity. Zero is not included in that situation. It does have a vertical asymptotodance, the x equals zero and a horizontal asymptotodate y equals zero. That part's the same. You go through the point one, one, which is the same, but you also go through the point negative one, one, which is different from the odd case. The magnitude will get, the graph gets steeper when you're near the origin and it gets flatter when you're away from the origin. That part was the same with the odd ones. The asymptotic behavior near the vertical asymptote is, well as x goes from the positive side, y would go to infinity, that part was the same. But as you approach negative infinity, excuse me, as you approach zero from the left, it's still positive, that differed. And then the end behavior is a little bit different as well. So basically when it comes to your odd degree functions, you basically get something like the following, your odd reciprocals. Essentially, again, this is the basic picture. You're gonna get to be something like this what you see here in blue. I mean, clearly this shouldn't be any sharp corners to be rounded, but it's gonna look like that. The power gets bigger, bigger, bigger. You wanna make it steeper and flatter. And this pivot point basically is around x equals one and x equals negative one. That's the idea for the odd case. For the even case, what we're saying is basically the following. Here's your asymptotes. For the even case, it basically looks like the following. This is the basic picture where again, there's no sharp corners here, but this will kind of pivot around the point x equals one and x equals negative one. You get this basic shape for the even reciprocal functions. This is in contrast to the even monomials which kind of looks like a bucket and the odd monomials which have this type of shape right here. So let me give you an example how we can graph irrational function using transformations. So if you have something like the following, let's graph the function r of x equals one over x minus two squared plus one. I want you to notice that fundamentally, we've just taken the graph y equals one over x squared and we transformed it. y equals one over x squared, that's one we were playing around with before. It'll look something like this. What are the transformations in play here? Well, the denominator here, this is our horizontal zone. Do, do, do, do, do, do. We're replacing x with x minus two. This will suggest there's a shift right by two units. And then, and so that's gonna take this picture and move it to the right by two units. So we go one, two. The point one, one will come over by two. So now it's gonna be the point right here at three, one. And then the point negative one, one will come over two units. So it actually becomes the point one, one. But this also moves the asymptotes. So the vertical asymptote at the y-axis, it'll come over by two. And it becomes now the axis x equals two. Notice that the horizontal asymptote's not affected by a horizontal shift. If a horizontal line shifted to the two, it's, it's, it didn't change, right? But that's what this minus two does inside right here. What does the plus one do? Well, the plus one is gonna shift the graph up by one. So everything goes up by one. So the point here, this point here at three, one will go up by one. It becomes the point now three comma two. This other point right here, it's at one, one. It's gonna shift up by one. So it now becomes the point one comma two as well. Your vertical asymptote, when you shift everything up by one, it's, it's still a vertical line. It's not gonna change. So your vertical asymptote stays at x equals two. Notice that when you look at this function, the problem with the domain is gonna be what makes the denominator go to zero, right? So you take x minus two squared equal to zero. That would imply x equal x minus two equals zero. Take the square root. So x equals two. So the problem with the domain coincides exactly with where this vertical asymptote ended up. But notice when you shift things upward that also will move the horizontal asymptote up by one. And so now we got a horizontal asymptote at y equals one. So using transformations, we can transform these reciprocal functions like so. And this is the first step to graphing rational functions. Now, in order to graph more complicated rational functions, we're gonna need some more advanced techniques, just like with the polynomial graphs we did before. The monomials, we could graph all of these monomial functions using transformations, but general polynomials, we need a little bit more. We need some more information. But graphing monomials was still important. The in behavior, for example, the behavior near a root, turns out that the polynomials were approximately monomial functions when you went towards infinity or when you got close to an x intercept. The same is also gonna be true for rational functions that as we go to the extremes, we go to the in behavior, or if we get close to an x intercept or we get close to a vertical asymptote, turns out our rational functions will be approximately the same thing as monomials or reciprocal functions. And that's why we need to know these graphs.