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Video transcript

in this video we're going to see if we can graph a rational function a raffle rational function is just a function that has an expression on the numerator and the denominator has a polynomial in the numerator let's say we have x squared over another polynomial in the denominator x squared minus 16 we could obviously graph this by just trying out a bunch of points and then connecting the dots that's what a calculator would do for us a graphing calculator but we want to do is before we try out some points to kind of fill in the gaps I would understand the basic structure of this graph first and to understand that I want to see what happens as X gets really big so X gets really big or X gets really really small as X goes in the negative direction or another way we could think about it I want to understand what happens as the magnitude of X or the absolute value of x becomes really big as it approaches really really big miss or as it approaches infinity so with the size of X approaches infinity which essentially is saying as X goes really far in the positive direction or X goes really far in the negative direction what is going to happen to the value of this function so let's get out a calculator we'll use the graphing part of it just yet but let's just try out some values what happens when X is equal to 10 and yeah it's going to be the same thing as when X is equal to negative 10 because when you put a negative 10 here you square it you get 100 just like 10 same your negative 10 you square it you get the same thing as a positive 10 so whether you go in the super high positive direction or the super low negative direction as you approach positive or negative infinity you're going to approach the same thing because you're squaring the values but let's try out some values if I get 10 squared divided by divided by 10 divided by 10 squared minus 16 I get 1 point 1 9 now what happens if X gets a little bit bigger this is X is equal to 10 what happens when X is equal to 100 we have 100 squared divided by divided by 100 squared minus sixteen I'm getting even closer to one this was what what it was X was ten when X is ten around here we're getting Y is one point one nine when X is one hundred one hundred squared over one hundred squared minus 16 y is one point zero zero one six this for fun let's try a thousand so it's one thousand squared divided by 1000 squared minus sixteen and we're even closer to one so as X as the size of X gets larger and larger and larger our Y gets closer and closer to one and that would also be true if this was a negative ten because negative ten squared over negative 10 squared minus 16 is going to be the exact same thing because the negative when you square it is going to be a positive it's going to be the same thing as 10 squared same thing over here so whether X gets really big or X gets really small we're going to be approaching Y is equal to one you could try it with a million if you want you're going to get a number even closer to one so it's the size of X approaches infinity the absolute value of x or the distance from the origin approaches infinity Y is approaching Y is approaching one or another way to think about it is the graph of this function is going to approach the line y is equal to one so let me graph the line y is equal to one so I'll do it in a dotted line because this is the graph of our function but this is a line that our function is approaching so that is the graph of y is equal to one now this idea of a function or the graph of a function approaching a line but never quite touching so this going to get closer and closer and closer to this line y equal to one in that direction but never quite getting too close close enough to it'll it'll get it'll approach zero its distance from this y equal one but never quite get there this line that the graph is approaching is called an asymptote as simple and it'll be even more clear once I actually graph the function we're going to work up there and since it's a horizontal line we call this a horizontal some tote horizontal asymptotes this is what our graph approaches as we go to the positive direction or really far in the negative direction let's think about some of the other interesting things about this about this function right here well one thing that might pop out at you is this is a difference of squares this is x squared minus 4 squared so we can rewrite this as x squared over X plus 4 times X minus 4 so what's going to happen here as X approaches either positive 4 or X approaches negative 4 well first of all try those values out if X is equal to 4 what is going to happen this expression right here this term right here is going to be equal to 0 and we're going to be dividing by 0 we cannot do that similarly if X is equal to negative 4 we'd be dividing by 0 this expression right here is going to be equal to 0 we can't we can't do that so this this we could say that this function function is undefined undefined at X is equal to plus or minus 4 it can't equal those values because we be dividing by 0 in either one of those circumstances now what happens as we approach those values what happens as X what happens as X approaches what happens as X approaches negative 4 let's just do that one for fun what happens as X approaches negative 4 and let's look at it from let's say let's say we're approaching it from the negative direction from the negative direction so let's try it out in our calculator let's try it out in the calculator so let's say we want to go from the negative direction so let's start with negative 4 point 1 so if we have if we have negative four point one negative four point one squared divided by divided by negative four point one squared minus 16 what do we get we get twenty point seven five so we get this number whatever let's see if we get even closer to negative four so let me just get that entry there so let's get a little bit closer to negative four so let me instead of four negative four point one let's do negative four point oh one so let me insert let me insert a negative so let me insert a negative four point oh one and then over here this is negative four point zero one four point oh one and see what it is now we went to 200 so we're getting to larger and larger values let's try four negative four point zero one let's try that out let's try that out whoops that's not what I wanted to do I wanted to do that so let's try no that's not what I want to do let's see so we want to go to instead of four point zero one I want to do four 0.001 and over there negative four point zero one four point zero one and what do we get we get two thousand so as we get closer and closer to negative four from the negative direction we're approaching larger and larger super larger numbers and you could try if it's for 0.0001 it's going to get to smaller and smaller numbers or sorry larger and larger numbers here you know if you do 4.00 one it's probably going to be a 20,000 and then if you add another zero here so as we get closer and closer it's getting to larger and larger numbers so we could say as X approaches negative four we could say Y is approaching infinity it's getting to a larger and larger and larger value but we can never quite get to X is equal to four it's undefined there that will make the denominator here equal to zero so we want to do here is we can never quite equal X equal negative four so let me see X is equal to one two three four we can never quite get to X is equal to negative four let me draw X is equal to negative four is a dotted line right there that is X is equal to negative four X is equal to negative or we can never quite get there but as we approached it from the negative side as we had you know four point one then four point oh one we went to larger and larger values and we also know that we want on the left hand side as we go to larger and larger X values that Y will get closer and closer to one so you have a general sense of what this part of the graph will look like this part of the graph is going to look something like that as X gets to super negative numbers it gets closer and closer to one as X gets closer and closer to negative four from the negative direction it's going to go closer and closer to infinity you're going to get closer and closer to a very it's going to get larger and larger I guess this is an easy way to say now just like X equal negative four x equals four will also be a point where the graph is undefined so let me graph that here one two three four right here right right over here X is equal to four and once again what happens is we approach x equals four let's say from the positive direction so as X approaches 4 from the positive direction what's going to happen so this is like trying out X is equal to 4.01 or X is equal to four point zero one or X is equal to four point zero one so we're just getting closer and closer and closer to X is equal to four well these values are the exact same values that we just tried on our calculator except they're the negative version of them right and we already saw that just the way that this function is set up the negative numbers they get squared so the whether you take the negative or the positive x values it's going to be the same thing this graph is symmetric when X is equal to negative 5 is the same thing as X is equal to 5 when X is equal to negative 10 is the same thing as x equals 10 so the same thing is going to happen you could try it out with your calculator if you like if you try out these values you're going to see as we get closer and closer to 4 we're going to approach larger and larger numbers these same numbers over here so the graph over here we're going to get we're going to as we get closer and closer to 4 we're going to approach larger and larger numbers and then here as X gets larger and larger larger we solve for here we had these horizontal happens to asymptote gets closer and closer to one so just like we called just as we called this a horizontal asymptote these values or you can even view these vertical lines X is equal to negative 4 and X is equal to 4 we call these vertical asymptotes vertical asymptotes simp Toad's these are lines asymptotes once again there are lines that the graph approaches but never quite touches so that's what's going on here and then we can think about what's happening what's happening to the graph inside of here so you could think of it in a couple of ways you could say well what's happening what happens as X approaches let's say what happens as X approaches 4 from the negative direction so let's try that out from the negative direction so what happens if you do 3.9 squared divided by 3 point 9 squared minus 16 you get negative 19 point 2 5 now what happens if we do 3 point 9 9 so let me put another 9 here so we're going to get closer and closer to 4 we're going to do it from the from from the left hand side as we approach 4 so insert another 9 here so even more negative so let's just do one more we're going to be even more negative let me make it 3 point 9 9 9 get even closer get even closer to 4 get even closer to 4 you're getting even more negative and this is also going to be true if we did negative 3 point 9 or negative 3 point 9 9 or negative 3 point 9 9 9 because that we when we squared the negatives and the positives are the same thing you square negative 1 you get a positive 1 so as we approach as we approach 4 from you know you go 3.9 3.9 9 we get closer and closer 4 we get large we get more and more negative numbers we approach negative infinity so as we approach let me just graph it here we're going to get as we approach from this direction we're going to smaller on not touch our asymptotes I'm just going to look at something like that as we approach it from the left-hand side reading smaller and smaller numbers and that's also going to be true as we approach negative 4 from the right hand side right as we get negative three point nine three point nine nine three point nine nine nine we're going to drop down it's going to look something like that and then we now that we have a general sense of what the graph is now is a good time where we could maybe plot a few points here so what happens and easiest one is what happens when X is equal to zero you have zero squared over zero squared minus 16 so that's so the point when X is equal to zero we're going to have 0 over well negative 16 which is just 0 so the point 0 0 the point 0 0 is on this curve and then we could try some other points if you like but the general shape here is going to look something like this it's going to look something like this you could you could plot more points if you really want to nail down exactly what the curve is doing in between but here is the general structure and you know we tried out a lot of values with the calculator and I did that because I really wanted to show you why it's dropping down like this and if you think about it it makes complete sense as you as you get closer and closer let's say you get closer and closer to 4 either way as you get closer and closer to 4 this is going to become a smaller and smaller and smaller number because this is the difference between x and 4 so if this has become a becoming a smaller and smaller and smaller number then when you take 1 over that right is you can eventually you could essentially view this as x over x squared over x plus 4 plus or times 1 over x minus 4 if this is becoming smaller and smaller this whole thing 1 over a super small number is a super large number so as you can imagine are going to get larger and larger and depending on whether you're approaching from the positive or negative so whether this is a whether this is a super small negative number or a super small positive number that's going to flip the sign but either way the magnitude so this is we're going to we're getting to a very large magnitude in the negative direction because the difference between x and 4 on this side is negative right 3.9 minus 4 is 0.1 take the inverse of that it's 10 so we're going to we're getting negative numbers here you take the inverse again super large negative number so I really want to give you that intuition but the general way of being able to graph these type of things your first thing you want to do is identify the horizontal asymptotes what happens is we get very low the magnitude of our X is very large so super positive values or super negative values you could try it out on a calculator if you like you literally if you try out the value million or a billion it will kind of give you the answer but the way you could also think about it is the other way to think about it is as X gets really large you could view that this thing these terms right here grow so much faster I mean this is just a constant term this term doesn't matter anymore if this is a million and a million who cares about the 16 so as X gets really large as X gets really large you could say that Y is approximately x squared over x squared these two terms dominate you don't need to worry about the 16 anymore and of course this is equal to 1 which is exactly what we got when we plugged in really large numbers so in a problem like this where you have the same coefficient or where this you have the same degree on the numerator and the denominator you look at the coefficient of those terms so in this case the coefficient is one and one so our horizontal asymptote is going to be 1 divided by 1 or Y is equal to 1 if this was 2 x squared over x squared minus 16 our horizontal asymptote would be y is equal to 2 we would approach that line up there if it was a negative 2 we our horizontal asymptote would be Y is equal to negative 2 so that's how you identify the horizontal asymptotes where you have the same degree in the numerator and the denominator and the denominator has a larger degree then the denominator is going to get larger much faster than the numerator and your asymptotes going to be zero I'll show an example of that in the future and obviously if your numerator has a higher degree than your denominator it's going to grow way faster than your denominator and you won't have any asymptotes you'll just keep growing or keep going in the negative direction and that's actually the case with all of the polynomials we've seen you could view them all as being over 1 in which case there was no no horizontal asymptotes now the vertical asymptotes you'd identify by essentially just factoring the denominator and figuring out where does it equal zero those are the points where the function is not defined and I'll show you in the future there are some special cases where they won't be vertical asymptotes and I guess that special case is for example if you had well I won't show you the special case right now I'll show you that in a future video but in general if you factor the bottom terms and they don't cancel out with anything on the numerator then you're going to be dealing with a vertical asymptote if I had another X minus 4 up here if my numerator was x squared times X minus 4 x squared times X minus 4 and then these cancelled out and my my expression simplified to this the equation would still be undefined at X is equal to 4 because you would give you a 0 in the denominator but since that X minus 4 cancels out with the X minus 4 in the numerator it would have not have been a vertical asymptote but that's and we will look at that in the future but this equation wasn't that so the general rule of thumb for identifying the vertical asymptotes factor the denominator figure out where the denominator equals 0 and if those terms don't cancel out with any terms in the numerator then those are vertical asymptotes and then to figure out the behavior I guess within the asymptotes you can plot some points you can try out some points you can actually you know substitute values for X and figure out what Y is now just to validate that we hopefully got the right answer let's actually graph let's actually graph our rational function so let me turn it on let me graph it let me say y is equal to x squared divided by X divided by x squared minus 16 and let's see what we get nope I just want to graph it well let me my range is off my range is off let me do my range and let me see X minimum value I want for X let's say it's negative 10 my maximum value I want for X is 10 X scale is 1 Y minimum value I want negative 10y maximum value I want 10 and then y-scale I want 1 now let me graph it there we go look at that just like what we drew we have an asymptote as we as X gets really large or X gets really really small that asymptote is y is equal to 1 we have our vertical asymptote it grafted because it tried to connect the dots but it really it essentially graphed our asymptotes for us but that wouldn't actually be part of the graph but as we approach 4 from 0 I guess we can say we go super negative as we approach negative 4 from 0 we get super negative because in either of those situations as we approach 4 from this side this term is going to be negative as we approach negative 4 from this side this term right here is going to be net well this term right here is going to be positive but then this term right here is going to be negative negative times a positive you could play with it does like we approach negative infinity in either case and then as X approaches infinity this thing asymptotes away so hopefully you found that fun