Graphing rational functions 1

right over here I have the graph of f of X and what I want to think about in this video is whether we could have sketched this graph just by looking at the definition of our function which is defined as a rational as a rational expression we have 2x plus 10 over 5x minus 15 so there's a couple of ways to do this first you might just want to pick out any numbers that are really easy to calculate so for example what happens when X is equal to zero so we could say F of 0 F of 0 is going to be equal to well all the X terms are going to be 0 so you're going to be left with 10 over negative 15 which is negative 10 15 which is negative 2/3 so you could plot that when x equals 0 f of x or y equals f of X is negative 2/3 and we see that point let me do that in a darker color you see that point right over there so we could have plotted that point we could have also said when does this function equal zero well the function is equal to zero when the only way to get the function equal to zero is if you get this numerator equal to zero so you could try to solve 2x plus 10 is equal to zero well that's going to happen when 2x is equal to negative 10 I just subtracted 10 from both sides and if I divide both sides by 2 that's going to happen when X is equal to negative 5 and you see that you see this right over here when X is equal to negative 5 the function intersects the x-axis so that's just two points but that still doesn't give us enough to really form this interesting shape over here and you could think about what other functions have this type of this type of shape so now what I want to think about is the behavior of the function at different points so first I want to think about when this function is undefined and what type of behavior we might expect for that function when it's undefined so this function is going to be undefined the only way I can think of to make this undefined is if I make the denominator equal to zero we don't know what it means to divide by zero that is undefined so this the function is going to be undefined when five x this in blue when five X minus 15 is equal to zero or adding 15 to both sides when five X is equal to 15 or dividing both sides by five five when X is equal to three f is undefined now there's a couple of ways for a function to be undefined at a point you could have something like this so let's let me draw some axes right over here so let's say that this is three you could have your function it could look like this it could be defined it might kind of approach something but just not be defined right at three and just keep on going like that or the other possibility is it might have a vertical asymptote there and if it has a vertical asymptote it's going to look something like this it might it might be approach it might just pop up to infinity and then it might pop down from infinity on this side or it might go from negative infinity right over here that says that's what a vertical asymptote would look like that as we approach from the left the graph is approaching a vertical but it never it never quite gets two x equals three I guess one we could say it or the function is defined not defined at x equals three and as you approach from the right the same thing the function just in this case drops down it almost becomes vertical its approaching negative infinity as X approaches three from the positive direction so how would we know obviously when you look at here when you look we know the graph ahead of time and if you say okay this is e this is x equals ten so let's see how many this is one two three four five so each of these are two so x equals three is right over here when you look at the graph if you had the graph in front of you you'd see oh look this is indeed a vertical asymptote just looking at the graph you see that you have a vertical asymptote at x equals three and let me write that down vertical asymptotes vertical asymptotes asymptote at x equals three at x equals three but how would you have known that how would you have known that if you didn't have the graph here if you just had this we know it's not defined at three but how do we know it's not a point discontinuity and not a instead of a vertical asymptote well there's a couple of ways to do it one ways you could try values near three and see what happens so for example you could get your calculator out and you could try let's say three point zero one so if you say two times three point zero one plus ten X G well that's the numerator and then I'm going to divide that the numerator by five times three point zero one minus fifteen it gets us a fairly large number it's kind of exploding on us and if we've got even closer so if we did two times three point zero zero one plus let me plus ten plus ten divided by five times three point zero zero one so now I'm just now I'm trying X is equal to three point zero zero one minus 15 minus 15 we see we get even a larger number so as X gets closer and closer to three f of X seems to be exploding it seems to be approaching positive infinity so that's one way to say okay this looks like at least from this side we are approaching we are approaching positive infinity so we would have been able to draw we would have been able to draw something like that and then you could have tried values below you could have tried values below so you could have said I actually let me just let me just put the last entry here and let me just change the 3.00 ones to two point nine nine nine two point nine nine nine and whoops let me go over here we have two point nine nine nine and we get notice we're going really negative now we're approaching negative infinity so if you just try that out that would give you a pretty good indication that the graph is looking something like that right over here which also seems to match connecting these two points that we've already thought about but now let's see what's going on as X approaches really large values or really negative really positive values are really negative values it looks like there is a horizontal asymptote here just looking at the graph just looking at the graph it seems like there's some value that as X approaches really large values really positive values f of X is going to be approaching that value that asymptotes from above and as X becomes really negative it looks like f of X is approaching that from below but how would we be able to figure that out just by looking at this well one thought experiment is just to say well what happens to f of X as X approaches infinity so let me write that down as X approaches infinity then then f of X is going to approach what well as X approaches larger and larger values the positive 10 and the negative 15 start to matter a lot less that the highest degree term in the numerator denominator start to dominate so we could say as X approaches infinity f of X is getting closer and closer to 2 x over 5 X which is which is 2/5 which is equal to 2/5 so you could say f of X is approaching 2/5 and if you really want to kind of see that a little bit more a little bit more concretely let's imagine different values for X as X gets larger and larger and larger so if we have so X f of X if X is 1 that f of X is just going to be 2 plus 10 over 5 minus 15 so here the 10 and the negative and the subtracting the 15 matter a lot but if X was if X were a thousand then f of X would be 2,000 plus 10 over 5,000 minus 15 so now the 2,000 and 5,000 are really kind of setting the agenda and then if X were if X were let's say 1 million 1 million and I just used blue for a more contrast then f of X would be 2 million 2 million plus 10 we move over to the right a little bit 2 million plus 10 over 5 million 5 million minus 15 so here the 10 and the 15 are almost inconsequential you could imagine if X were a billion or trillion or googol then these then the 10 and the negative 15 starting to matter a lot and lot less so as X approaches infinity these matter less the higher degree terms the degree terms matter so f of X is going to approach 2x over 5x which is 2/5 so f of X is approaching 2/5 and that's what this line looks like 2/5 is the same thing as 0.4 so f of X and we see that in the graph f of X is approaching that but it's not quite getting close to it it's it was it's not quite getting there it's getting closer and closer to it as X goes to infinity but it's not quite getting there because you're always going to have that plus 10 and that minus 15 there's you're never going to be exactly at 2/5 and the same thing is happening as X gets more and more and more negative you can make all of these negative values so if if this were negative 1 then it would be negative 2 negative 5 this was negative 1000 would be negative 2000 over negative 5000 if there were negative 1 million it would be negative 2 million over negative 5 million but you see even in this case f of X is approaching 2x over 5x which is approaching 2/5 or you could say it's approaching negative 2 over negative 5 which is still 2/5 and you see that right over here so we would say that this function has a horizontal asymptote horizontal horizontal asymptote asymptote at y equals the horizontal line right over here is y is equal to 2/5 so hopefully this graph here is helping us appreciate what these vertical and horizontal asymptotes actually are but if we didn't have the graph we could have said okay we're undefined at x equals 3 we could test some values around it and so we could say okay look it does look like we're approaching negative infinity as X approaches negative 3 from the left it looks like we're approaching positive infinity as X approaches negative 3 from the right so we could we could put we could draw that kind of blue point right there we could graph these two points when does F equal 0 what happens to F when x equals 0 and then we could think about the behavior as X approaches infinity or negative infinity as X approaches negative as X approaches infinity or negative infinity and draw this horizontal asymptote and between all of those that would have been a pretty good way to be able to sketch this actual graph