 So our goal for the last several videos has been able to develop a technique for which we can graph various functions without the need of a graphing utility of any kind and It works really great when we have graph transformations and we know a basic graph. We can shift it We can stretch it. We can compress it. We can reflect it. We can handle those things fairly well by now But there are some functions that are a lot more complicated for which transformations are insufficient for our purposes I take the function g of x equals x to the two-thirds times x minus two to the one-third This function here, we're not going to be able to graph it via transformations alone now in previous videos for this lecture 48 we talked about the idea of in behavior and how dominant terms come into play to determine the in behavior Acentotically our function could look like an exponential. It could look like a power function or something in that in that direction And so dominance that helps us know that that as we go to the far left and the far right of the graph Then our great our functional asymptotic look like a simple function like a power function or exponential What have you but we don't have any information yet about what happens in the middle If we had a function like y equals say x squared times x minus two something like this We actually had some experience with this like okay Acentotically, this is the same thing as x cubed great But we also news things about the x intercepts right you have an x intercept at x equals zero You have an x intercept at two an x equals zero will touch infinity and at x equals two little excuse me at x squared because of it because of the x squared the multiplicity will will cross the x-axis at Two and will touch the x-axis at zero right there So you get a graph that looks something like the following right? So we have the asymptotic behavior of x cubed it goes up on the right it goes down on the left But then in the middle near the x intercepts we know whether it touches or crosses so we can use in addition to the in behavior We can use the asymptotics of the intercepts. That is what happens near the x intercepts Right and so as we studied this for polynomial graphs Say we have a polynomial function like the following f of x equals x minus the first root to the m1 Times x minus r2 the second root to m2 the second multiplicity all the way down to the nth root x minus rn to the mn And so each of these cases r1 r2 rn These are the roots of the polynomial the x intercepts of the graph and then these m's here We're the multiplicities right and so the strategy we mentioned earlier is that Asymptotically as x approaches its x intercept here that is as x approaches r i the function will be approximately f of x will be approximately it'll look like c times x minus r i to the mi for some Coefficient c now. How do you find that coefficient? Well, you're just gonna plug in x equals r i for all of the x's With one exception you don't plug it in for the eyes for the r i spot because otherwise you get zero So you plug in x equals zero everywhere else and then you can see that our function will look approximately the same thing as a shifted Power function in this in this regard some time to try to train some transform power function for which we can then We can then graph those type of things So in other words as x is closer and closer to an intercept the graph will closely resemble these power functions with transformations at play for which the multiplicity will help us determine How does the power function behave right if it was even you touch the x-axis Buton cross if it was odd you would cross the x-axis so that we've learned about that before With reciprocal functions, there could be vertical asymptotes in play like if you're trying to graph y equals let's say x minus one to the negative one power that could be rewritten as One over x minus one and so it's like oh that looks like the reciprocal graph where you have x in have a vertical asymptote It would look something like this right, but you've shifted it to the right by one We could do those ones as well What I want to focus on in this video because we've done that with we've done that with the reciprocal functions in the monomial functions What happens if you have like a fractional exponent like two-thirds and one-third well then we have to figure out well What does the graph y equals x to the say a over b power? What does that look like in general for a fraction? Okay? Let's say your multiplicity m turned out to be a fraction a b well The first thing to do is you want to write that fraction in lowest terms if a and b have any common divisors Crush them take them out and simplify it. So if you had something like two-fourths, you're gonna replace it with one half instead Okay, put it in lowest terms So when the numerator a is odd, right? When a is odd that means that your graph is going to cross the x-axis at that route Just like we saw with monomial functions you cross the x-axis with the reciprocal functions We said that we would cross infinity our spaceship would fly off the screen and wrap around from the other side When a is an even number the graph is going to touch the x-axis at that route We saw that when we had even monomials like x squared We also saw that when you had like even Reciprocals like one over x squared that you have your vertical acetone you would touch infinity So you touch at the same side and instead of wrapping around the other way So we've seen that before so if the numerators odd you cross if the numerator is even you touch that pattern is consistent With what we've already seen before okay, but It turns out there is another thing we have to pay attention to we have to look at in itself When m is a big number that is it's bigger than one Right so that could be it like it's like two three or four like a like an integer But we also take something like five halves or say six fifths or you know seven third things like that We could still have a fraction that's bigger than one if your fraction is bigger than one What that tells us is that the graph will become flat near the route Okay, and the concavity it'll concave away from the x from the x-axis in that regard So what I mean is when you're when your m is bigger than one It's gonna be kind of like the following if you're odd you might cross and you get something like this So it's flat near the x-intercept and the concavity is going away from the x-intercept from the x-intercept If it was an even graph though you might get something like the following right it gets flat It gets flat near the x-intercept and concavity goes away from the x-axis there and then the bigger m is the flatter it gets Right, it's the flatter The bigger m is okay, so that's that's pretty nice now Let's say that your your multiplicity is between zero and one and I mean it could be negative But if it's negative it's not an x-intercept as a vertical asymptote So you do everything Accordingly that way so we're just focused on the absolute value here if the absolute value here of m is between zero and one Small number let let's say leg was like one half or one third or maybe it was like three fourths something small Right's less than one the graph then instead will become steep near the root So it's gonna be really really steep that is The graph will concave towards the x-axis when the graph crosses the x-axis steeply We say that the graph has unusual steepness, right? It's gonna be very very very steep Okay, and so one possibility to get something like the following. Let's say that your Let's say that you had your your fraction here m a over b. We're gonna have that a is odd A is odd But m is less than one here and that situation you would expect that your graph is going to be steep Right, we want we don't want it to be flat like these pictures. We had before we want it to be steep Right, so we get something like this So the their steepness, right? It's when you look really point the point to get steep And then the closer and closer you get to zero the more steep it becomes notice the concavity in this situation The concavity is pointing towards the x-axis. Okay, that's if you're that's if you're Odd you have this unusual steepness, right? On the other hand if you're even if you're even you get the same picture Right that it should be it should be very steep as you get towards the x-intercept the concavity should be pointing down But you're only touching the x-axis you can't cross it So you have to get something like the following right for which it's steep near the intercept But it should be concave down towards that is concave towards the x-axis in this situation your unusual steepness actually looks like a cusp Right because you get this thing. Let me kind of remove the x-axis there. Whoops I'll just draw it a little bit higher. You get this like sharp corner right here It's a cusp so as you're looking through homework problems They often refer to this even be steepness that this is called a cusp But if it's odd they might refer to it as like unusual steepness It's really really steep as you get close to the x-intercept and so that with that difference in mind We're now ready to start graphing functions that kind of look like polynomials But we might have fractional powers in play here. So this example we have G of x equals x to the two-thirds times x minus two to the one-third so things I can detect very quickly So as we look at this one We're going to get an x-intercept at zero and we have a multiplicity of two-thirds some things to notice here about M M is less than one right and The numerator is even Right the numerators even here that means we're going to touch the x-axis We touch the x-axis But because M is less than one this needs to be steep right so the end that tells us our graph is going to have a cusp at x equals zero Alright, let's look at the other factor if we look at x minus two to the one-third power that tells me that I'm having an x-intercept at two Okay, the multiplicity there is one-third, which you'll notice one-third is also less than one So that tells us that this thing is likewise going to be steep but Because the numerator is odd We see that it's going to cross It's going to cross the x-axis and so we're going to refer to this as unusual steepness Unusually steep Unusually steep that's that's how we're going to describe this thing So that's going to be the behavior near the x-intercepts. What is the what's the asymptotic behavior? Notice if we only look at the leading terms we get x to the two-thirds and we get x to the one-third Asymptotically as we go towards infinity or negative infinity This will look like two x to two-thirds times x to the one-third which adding the powers together You get x to the three-thirds which is equal to x itself And so this tells me that our function will be asymptotic to a line That is there's going to be an oblique asymptote Oblique an oblique asymptote at y equals x Okay, and so let's put this information together Right. I'm going to have the the finished picture on the left-hand side And then we're going to draw it on the right-hand side here to kind of show you what's happening So if we put our x-axis here and our y-axis here, I'm not going to draw this thing perfectly to scale But what we anticipate is like we said there is an x-intercept at zero and an x-intercept at two at and then We have this this asymptote this asymptotic behavior at y equals x, which is this diagonal line That looks something like the following Right. We have this asymptotic behavior We might actually want to figure out whether we cross the asymptote or not I'm not going to worry about in that in this situation though. Let's just go with the information We have so that we know that at x equals zero There's going to be a cusp. So the question is is going to be a cusp like this or is it going to be a cusp like this? We see on the picture that should be down on the downside, but how do we know that right? We'll come back to that in a second We also know that at We also know that at x equals two there should be unusual steepness So should either look like this like the picture suggests or should go the other way around How do we know which one is which well, like I said, we could figure out when does our function? Does it ever cross our horizontal asymptote or not? Another approach is actually just to plug into our function, right? We have our function f of x equals remember x the two-thirds times x minus two to the one-third What happens if I plug in? You know what happens when x approaches zero right here? Well, what this means is you're gonna get f here will be approximately the same thing It'll be approximately the same thing as x to the two-thirds times Zero minus two to the one-third like so so this thing will be the same thing as negative the cube root of two times X to the two-thirds so the the reason this matters here is the sign, right? This has a negative coefficient some means instead of looking like x to the two-thirds which has that cusp It's been reflected down. So our cusp is going to go downward. That's what we care about right there So having x approach zero tells us that it should be reflected downwards So the cusp should be pointing up like we see right here Okay, so that honestly is enough information to finish the rest of the picture Because we can see that from here I'm gonna have to come towards my my my x intercept passing through it like that and then you're gonna approach your asymptote after that But if you wanted to you could do it at the other x intercept as well as x approaches to what happens Well, our function will be asymptotically the same thing as two to the two-thirds times X minus two to the one-third like so you'll notice in this situation that the coefficient is positive That's all that matters We want to know is there a reflection that we reflect across the x-axis since it's a positive coefficient There's no reflection so it should look like the standard function which looks like this the reflected form would look something like that So we then get our picture For this one right here. Let's look at another example in this example Let's consider the function g of x equals the square root of x times x minus two times x plus three Now because of the square root here It might be a little bit misleading But really what we want to do is rewrite this in the following way rewrite this as x to the one-half times x minus two to the one-half times x plus three to the one-half right all these situations and so this tells us very quickly Okay, we have an x intercept at zero. We have an x intercept at two. We have an x intercept at negative three okay We then our multiplicities are going to look like one-half in each of those situations one-half one-half one-half Which one-half is the numerator is one, but it's less than the multiplicity one-half is less than one So this indicates to us that all of these ones should have this unusual steepness unusual steepness Near the rex intercepts But there's also another thing we need to draw our attention to We have a square root Right and we have variables inside of square root. There could be some choices of x for which That we're taking the square of a negative. Ah, okay, so we have to actually think about it for a second What's the domain of this thing? So with this then comes down to solving the inequality x times x minus two times x plus three We need this to be greater than or equal to zero Okay, which that leads ourself to a number line for which we label the number line with the same markers zero two and negative three Okay, for which we then see here that we get this If we if we think of the graph of this polynomial So forget all of the powers of one-half from on if you think of the polynomial each of these things have odd multiplicity one one one, right? And this is approximately the same thing as an x cubed So it should be pointing up on the right-hand side It should be pointing down on the left-hand side and then we cross the x-axis all the time So we're gonna get something like this notice that we're negative here positive here negative here and positive here this tells us that the domain The domain of our function g is going to be the interval negative three to zero union Two to infinity. Okay, so in terms of in behavior We actually are interested in what happens as we get close to infinity What happens when we get close to two zero and negative three? That's what we really should be worried about here Okay, but we can answer all of those questions so that as x approaches infinity We're gonna look like the dominant term This is gonna look like y equals the square root of x times x times x that is this will look like x to the three-halves In particular this is gonna go off towards infinity as we approach to let me switch to a different color as we approach to Our function is gonna look like the square root of two times x minus two times by three Excuse me by five Which is two plus two plus three right there for which then we can see that okay This is gonna look like the square root of ten times x minus x minus two to the one-half And so in particular it's gonna look like a square root function. So it's going to be Exiting x equals two from the positive side. And so if we put that into practice here Okay, let's let's kind of graph the information we have so far if we were to graph this by hand Let's make that look a little bit straighter We get this and we get this right here So what we're saying here is that When you're at x equals two it should look like a standard square root So we it should be coming up and then the other one says it should be approaching x equals three-halves for which it's Steeper than a line, but it's not as steep as a parabola. So we get something that looks like the following Exactly what we see here from the computer generated picture, okay Now what is it? How does it behave near the other intercepts? Okay, because it only there's only gonna be a graph that we only have anything between negative three and zero Everything else is outside the graph So if you look at towards negative three, we're gonna plug negative three into all the pieces except for X plus three for which case you're gonna get negative three times negative five times X Mine x plus three right here for which this is gonna look like the square root of 15 Times x plus three to the one half power for which again that thing should look like a standard square root Kind of coming up like so and then the last one if we consider that one as well Then we plug in zero everywhere except for the x. So we're gonna get the square root of x times in this situation we end up with We have a zero minus two so it's gonna be a negative two right here, and then we're gonna have Then we're gonna have a X plus three which the x became a zero so we're gonna go three right here So this thing is gonna look like the square root of six Times in this case the square root of negative x right now this this changes the domain right this thing because of the negative sign here It's reflected to across in the other direction So this should actually look like a picture looks something like this for which the other one looks something like this And so when we put that together as we put this together We should go from zero to negative three it should come up and then come back down right something like that now That's not exactly what our picture has notice. That's a little bit more lopsided towards one side or the other We're not gonna worry about where that maximum point is right now. We're just trying to get this picture that we see right here Okay, let's do one more example Let's do one more example of this thing right here h of x equals x over the cube root of x squared minus 4 So what can be going on here? What do we should be paying attention to? Well, if we think about the domain right the domain should be of concern to us You'll notice that the denominator you're gonna get the cube root of x minus 2 times the cube root of x plus 2 if we factor the denominator and then the numerator we have an x So some things we could ask ourselves is in terms of domain. Well, since you have a cube root There's no worry about plus versus minus, but we do it. We are divided by zero at certain places, right? So we're gonna have we're gonna have vertical asymptotes at x equals 2 and at x equals negative 2 right and the multiplicities of those things are gonna be 1 3rd Right and equals 1 3rd because of the cube root there. We also have x intercepts And there's gonna be an x intercept at x equals 0 and its multiplicity is gonna equal 1 and then the last thing I'd be kind of interested in is the in behavior, right What's the in behavior of this thing because this thing will this this function here will be approximately the dominant term On top is gonna be an x the dominant term on the bottom is gonna be a cube root of x squared That is it's gonna look like x to the three halves For which as x approaches infinity, I guess we should reduce that thing first down We're gonna get 1 over x to the one half and so as x approaches infinity right as x approaches infinity we see that We're gonna get 1 over 1 over what did I write? What did I write I have I have my fraction upside down. I'm sorry That was a mistake on my part. Let me fix that so you get x over two-thirds For which then when you subtract one from that that makes a big difference On how this thing looks right if you take if you take x the first minus x to the two third That is minus the powers there. You're actually gonna get x the one third like so And so in this situation as x approaches infinity we see that Y will approach infinity and then as x approaches negative infinity We see that x the one third will approach negative infinity as well. Okay, and In particular our function will be asymptotic to this value right here That that is this function will look approximately like this It'll look approximately like this x equals one third there Okay, so with that information I think we're in a situation where we can start to look at what this graph is going to look like This is the final picture right, but I want to first Let's try to graph it on our own and see what's going on here. So remember this picture will come back to it in just a second Okay, let's graph thing. I'm gonna Emphasize who's the x-axis and the y-axis so we can see that here I'll do something like this. So what information did we discover right? So we have an x intercept at zero Okay, so let's mention that point We have an x intercept at zero like so let me do that with yellow So it sticks out a little bit more we had a vertical asymptote at two and negative two So add those onto the picture like so And then something like this great What else did we know about these things? So if we mentioned like their multiplicities, right? The multiplicity of the x intercept was m equals one So this will and then and then the multiplicities of the of the asymptotes This makes this was one third remember So what this why this matters for us is that we're going to cross infinity at this asymptote We're going to cross infinity at this asymptote because the numerator here is odd We're also going to cross We're going to cross the x-axis at our intercept. So that that part we know we also know the in behavior that We don't want in behavior that as we go off towards Infinity y will go towards infinity and as we go towards a negative infinity We're going to go off towards a negative infinity And so because of this information we can actually have enough to start piecing things together Because if we start off in this this area over here, we know we have to go up towards We don't have to go off towards infinity as we approach the asymptote We have to have to go up or we go down, right? But there's no x intercept right here. So we can't go from over here to over here So what's going to happen is we're going to have to get something that looks like the following Right at some point it's going to come and bend around like so going off towards Infinity on the right hand side and we'll approach infinity as x goes towards Two from the right hand side. Okay, but now we know we cross infinity So we have to come up from the other side of the picture. Okay Then we have to approach our intercept for which at the intercept We're going to cross the x-axis come over here. And so then as we get towards our Vertical asymptote again, we either go off towards infinity or we go off towards negative infinity. Those are the options But if we're going to go towards negative infinity, this would require us crossing the x-axis Which we can't do that. So we're going to have to go off towards Infinity in this approach again, since we cross the horizontal Since we cross the vertical asymptote, we're going to have to wrap around from the other side And then we have to get towards negative infinity, right? We can't go too far up We're going to have to wrap around and do something like this All right, and so that then gives us our picture when we use the multiplicities of our intercepts and our vertical asymptotes here So let's try to compare these can I put these on the same on the same picture right there You can see our hand drawn one versus the computer animated one. And yeah, it's pretty good We're able to find all of this information using the asymptotics of our graph