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2.6 Rational Functions & Asymptotes

Added Nov 05, 2015 •

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There is an error in the very last problem. The function should have been g(x)=(-x^2-2x+8)/(x-2). I have a +6 instead of a +8. That's what happens when making videos late at night! :)

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high class today we're going to work on Section 2.6 so we're skipping ahead of chapter and a few sections and we'll be looking at rational functions and asymptotes so go ahead and get your spiral notebook and copy out the objective for today the objective is I will identify asymptotes horizontal asymptotes and holes in rational functions so you might need to pause this for a minute while everybody gets that copied out turn the first thing we're going to need to look at is what is a rational function so a rational function comes in the form of fxrg avexir age of x equals x over DFX and into stands for some numerator over a denominator so its expression is going to be written as a fraction so he can kind of think of rational functions as functions that are written as fractions  now one of the most common rational functions is f of x equals 1 over X it's kind of like the parent function of rational functions if you compare it to what we've been studying recently so  the graph of 1 over X looks like this on Desmos and just some things to point out about scrap that we've talked about earlier we've been talking about domain domain is your X values how far to the left it goes and have for the right of a ghost so this say crack has a domain from negative Infinity all the way to zero and then you have to jump over the zero and then it goes from go all the way to Infinity it's raining just how low does it go so it goes all the way down to negative Infinity up to 0 again and then you're going to have a gap and Siri got to jump over it so be nothing to do zero and then 02 Infiniti  another can we've talked about recently is whether this functional be even or neither will be even if you could fold in half on the y-axis and if it was on top of each other with symmetric with respect to the y-axis and it would be even this is not the case here you couldn't fold it in half on the y-axis and have it be on top it's odd if it's too much as with respect to the origin so if we take this piece right here and fold it in half on the y-axis it would be here and then fold it in half on the x-axis then it would be down here so that in fact is too much of quizzes but to the origins of this is an odd function do those are few things that we've talked about previously and now we're going to add a few more characteristics here where we look at Classen tops and horizontal asymptotes  so on this next slide I don't need you to write this down I just want you to look at it and think about it I hear we've got to assess the line x equals a is a vertical asymptote of the graph if f of x approaches infinity or f of x approaches negative Infinity as X approaches a either from the make a from the list so that sounds kind of complicated let's go back to the line x equals a is a vertical asymptote vertical line so in this particular graph I'm just going to tell you that the vertical asymptote is  VA Nicole asymptote is x equals 0 to make a graph would represent with Lions so I have a vertical asymptote here at x equals 0 all along the y-axis  if you know that that's your vertical as a no and I just told you that it was you'll have to trust me for a minute but if I told you that tell you that goes over to class and Pope let's see if this makes sense so says if which is my wife is approaching Infinity I have a vertical asymptote as X approaches a well here in this case my a is zero this is my equals a is 0-2 as I put zero from this side I'm going 5 4 3 2 1 0 turn up routine 0 where is Minecraft going it's going up and up and up and up and up to Infinity  now if I go on the other side and I posted from this side is going down down down dun dun dun dun down to negative Infinity so that's what this definition says up here it says that if your either positive infinity or negative Infinity as X approaches a then that is your vertical ass the same is true of a horizontal line only its flipped around a little bit so far horizontal and I'm going to tell you that in this particular graph my horizontal asymptote is y equals 0  turn the make a dotted line along why member why always makes a horizontal line because why is horizontal Okay so knowing that we have a horizontal asymptote why should breviate that with each day here horizontal asymptote horizontal asymptote  okay so if I look at this white was be something my B 0 is a horizontal asymptote if my y approach has be as X approaches Infinity 18 of this is why is approaching zero as X is approaching Infinity well let's see out here as X is going to Infinity play out here What's My Line are you my Why by you is getting close to zero and same with this one as X approaches negative is way out here as X is approaching naked and Affinity my wife is approaching zero so my horizontal asymptote is not any of the stuff down we'll look at it more as some more examples right now this light I do want you to write down actually I don't want that that yet caso4  vertical asymptotes vertical asymptotes are always act what's something go ahead and write this out for vertical asymptote set the denominator and then you solve for x what for example we've got f of x equals 2 / x - 3 tell if I want you to find the vertical asymptote then we're going to take our denominator x -3 and said it equal to zero and then solve for x equals 3 say vertical asymptote at x equals 3  now let's take a look at what that looks like graphically I just said that you got a vertical asymptote at x equals 3 so if I went over 1 2 3 on this crap now you don't need to drive this crap that you can just dial this picture here next week today you just have to find them you don't actually have to grab them  so we have a vertical asymptote on this crap and it's true as I get closer and closer to this 3 sign up to infinity and I get as I get closer and closer this time going down to negative Infinity so you can see if I look at the graph that are results are confirmed x equals 3 does look like it will be a vertical asymptote it's not touch it tell us look at a few more examples 3/2 a 5 over x squared minus x minus 6 so go ahead and write this one down  and I want you to find the vertical help  you need to take your denominator equal factor that see if you factored it correctly should factor into -3 + 2 and means word have vertical asymptotes at 3 and -2 there's a vertical asymptotes now let's go ahead and confirm that with a graph and again you don't have to draw the graph out this is just for you to get a v understanding of what's going on here so using Desmos again I graft 3 + -2  you going to get a worksheet later if you want to use Desmos or your graphing calculators to graph the functions on your worksheet at just to see what they look like you can but it's not NBA choir today  so we've got x equals 3 is a vertical asymptote so if you go into your this crap over here see we've got a vertical asymptote right about there it looks like those purple lions and then at -2 playing around there so our graph gets closer and closer and closer to the x equals 3 here and over here is gets closer and closer to the x equals negative to both up there and down here as I get closer I'm going to down to negative Infinity here and after positive Infinity on each of those pieces that's not the vertical asymptotes next we have horizontal asymptotes  turn right these out so horizontal asymptotes there's three situations  I want you to ride out to Big Top fake bottom and set the same so an example of the use of three different situations that you're going to have our first one for a big toe you might have something like x squared over x + 1 a big bottom play b x + 2 / x squared + 3  and the same way next plus 3x out there and the same would be f of x equals 4x + 1 / 2 x - 6 now the reason why we say that it's Big Top at Lee bottom is because we're looking at the exponents  the largest exponent in both the numerator and the denominator so here this is X to the second here this would be next to the first so the exponent on top is larger than the exponent in the denominator so you have a big top now this one here for the you've got your denominator is X to the second where's your new murders only x to the first two again it's bigger on the bottom is there a singing this one's next to the first and it's next to the first so he say that those are the same so they're going to have and we've got y equals see if you have a big toe horizontal axis horizontal asymptotes are always y equals something where vertical ass and pups are always asking for something so for a big top is why you have no horizontal asymptote if you have a big top and you don't need to drive the graph out again but listen example of a graph that has no horizontal asymptote it actually has a has a vertical asymptote this one does and this one has something that's going to be called a slant asymptotes that will get into next week but to be a horizontal asymptote you'd have to have a horizontal line with a graph is getting close and there is no horizontal line where the this particular graph is getting close to so and that will always be the case if you have a big not have a horizontal asymptote set for big bottom fig bottomless always wife was 0 and this particular graph if I graph this one out on Desmos now this is kind of don't you think I know it's a little bit hard to see who me outline it so we can see a little bit better  so this particular graphics like  felt-lined it okay it's something like that I do so but you get the idea so this one says we've got a horizontal asset about white was there to find me and my daughter asked some top right here the interesting thing about this is as an approaching Infinity I am getting you can see that this crap is getting closer and closer and closer to fasten temp and same thing here I go to negative Infinity the grass was getting closer and closer so why Cozier is definitely a horizontal asymptote based on the death Santa as we go to positive infinity or negative Infinity we're approaching whatever that happened took is however it's a little strange here cuz this is special case normally your grass don't touch your asymptote but this particular one does in the middle here and that's okay it just often are graphs don't cross to the ass and put but sometimes with special in the middle piece to our graph sometimes it will cost through the isotope so just be where have the special situations they do exist  and then let's look at our third and final case is when the coefficient of so again we've got a next to the first and annexed to the first so if that happens then what you do is you divide your coefficients  so you take the number in front of the X which is A4 and the number in front of the X which is to can you just buy them so forward to or why was too so are horizontal asymptotes would be why question a graph with that ho function looks like a graph that out on Desmos and getting since a little harder to see to Let Me Trace that will cook for you so can stand up canh-sat  and then the horizontal asymptote it says is y equals 2 my little dotted asymptote there and hopefully you can see that the graph on the right hand side he was getting close and closer to that asymptote but not touching it on the graph on the left it's getting closer and closer to that acetone  set three cases Big Top none big bottom zero saying play coefficients final thing we have to look at actually  happy Friday. I have some examples for you to try and so go ahead write these three examples out okay so take a minute pause the video right here somebody can get these copied out and then I want if you can find just the horizontal asymptote in all three of these functions okay so I'm assuming you've had a go at it and have tried to find a horizontal asymptote Benito these functions so let's go and see how you so the first one you should look at your largest exponents you've got you've got a shoe over a three so it's a big bottom so big bottom is y equals 0 I always refer back to those notes if you need them and the next one we've got two two two  so Scott cider the same position so why equals  you going to have to divide your coefficients so you going to have 6 / 12 say your horizontal asymptote will be like what's 1/2 cast you got a 4 / 8:30 so that's a big set the top news for you got those right and we're going to go ahead and look at our third and final situation which is holes  so you have holes in graph sometimes  the easiest way from me to explain the is to just look at a situation in which you have a hole so go ahead and copy this function out f of x equals 3 * x + 1 / x + 1 time next -2  so there's a function for us and we're going to be asked to see if this function has any holes in the graph and it does because when you have matching factors here when they cancel off A co-worker there so the first thing I always have to do if you have a common factors is to simplifying so counting cancel this off so to find your whole you take your factor that cancelled off set an equal to 0  sewer hose actually going to occur at x equals one so we have a whole lot exical or sex -1x + -1 now what's that mean so let's look at the graph of it if we skip this out I know I have a vertical asymptote well my function  if I go up here eclipse occur when your denominator equal 0 and I crossed the Sci-Fi simplified at first so I don't worry about the sex plus one anymore what's on my denominator that I have left is just x -2 so that's when my denominator equal 0 so that means my vertical asymptote is going to be too now if I go and I sketch horizontal my horizontal asymptote chamber X-Men goodbye I'm in no x's on the top that's a big bottom so big bottom is y equals 0 so why koziro gives me a horizontal asymptote right along here aynax equals to give me a vertical asymptote my darling here hey we still haven't discussed what happens with this here we need to go ahead and grass first so I need to know to graph this rational function I need to plot some extra points if I plug in three to my function I'm going to have 3 / 3 - to do much to as one so 3/1 is 3 so we have a point at 3:30 Savannah Pointe at three 3s find another point if I plug in a stand on founded by the S&P plane that have a plane at 3:30 then I know my graph is going to be over here in this piece but I need to figure out where is it over here sit over here sometime so I'm going to plug in a point maybe like 1 so I can figure out what's happening on this side so if I plug in one  say that give me 1-2 is -1 - 3 over negative one to be negative 3 7 go down 3 here I didn't make my lines very evenly spaced LOL  so n0 I get -3/2 another points I'm going to go ahead and up with a curve like that so that's what this funk turn looks like except I still haven't discussed this whole so at night with one I have a horse I need to go ahead and graph negative one so there's negative one but there's a hole there doesn't look like there's a hole there right now so look carefully everybody looking stare right there the hands disappear again so we have a whole there so you have to actually make a little hole in your graph when X is -1 and then I just make an open circle so you can see it nicely like that where is it in this case it was at -1 and you erase that little piece and you put a hole there  okay let's try some exam false so go ahead and write doubt set three functions for you to practice on so go out and paws radio right now everybody get some coffee down and then go ahead and just find the holes in those three functions I'm assuming you have had time now to find the holes in each of those functions so let's see if you've done so correctly tell the x - 3 is cancelled off so you should have had a hole at x equals 3  in the second example the ex- once crossed off so you'd have a hole at exit also the X's cancel there and when they cancel you get a hold at x equals 0 be careful of those X's at front and this last one that did nominator here was not factored for you so that's the case we're going to have two factors so we need to factor that x - 5 + x - 2 what is x equals so hopefully you got this right one finals putting it all together  so now what I'm going to ask you to do is to find the vertical asymptote horizontal asymptotes and holes in the functions below and I have two functions for you to write out  so go ahead so I want you to go ahead pause this for a moment  can see if you can find the vertical horizontal asymptotes horizontal asymptotes and holes in both of those functions okay I'm assuming you've had some time to try that now so let's see how you do the class and hope it is when your denominator equals here but first you're going to have to factor that  factors nice me and the ex plus ones canceled so when you set that x -3 your denominator equal to zero should have had x equals three  horse I have some truth see if you have one over x minus 3 after you cancelled off the X One Source at Big Top big bottom of the same well it's a text to the first and nothing at no x's on top sensei big bottom vitamin spikeless hero and our whole occurs at x equals where they cancelled so the X plus one equals zero or x equals negative one are their function is a little bit trickier to factor you had to factor out an x on top so if you factor out an X all the when's change and then that factor some more you can factor that down into x + 4 + x - 2 honor x -2  6 planets to cancel  and so that means my vertical asymptote will see the verdict awesome trophies when I set the denominator equal this here but I just cancelled it off so I have no denominator so when I have to go down there I have nothing to say equal to zero so this is kind of an interesting case so my vertical asymptote is actually none there is no vertical asymptote since there is no denominator  iqor Lassen tow pussy big topic bottom of the same well I'll in love with his Negative X plus 4 I guess I could put that over one so this play Big Top because his ex to the first over to store one so if it's a big top it's y equals and then last my whole will holes are where you cancel them off so because we cancelled off x -2 you said good equal to zero and you'll get x equals a positive tail  that concludes our lesson for the day so you're going to receive a worksheet now with some belief 10 problems on it and you can go ahead and get started on that and that will be due on Monday so a great day everybody

Uploaded by Kristin DeWit.

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2.6 Rational Functions & Asymptotes

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