okay hey everyone and welcome uh this is mrs haney and today we're going to be covering 103 rational functions so this is our third lesson already in this semester can you guys believe it we are just zooming along are we not all right so let's take a look what we got all right so these are our objectives for today uh what we're going to be looking at and learning about we're going to be talking about rational functions we're going to start out easy we're just going to define a rational function we're going to see if we can do some examples of it we're going to talk about the domain and the range of rational functions we're going to talk about key features of a rational function on their graph and we're going to talk about real world applications if you'll notice as you go along even on your assignments that are just multiple choice there is always going to be one real world connection which is nice because a lot of students always ask well why are we learning this and this gives you an example of why we are learning it right anyone have any idea what a rational function is raise your hand if you think you know what a rational function is you ever heard of those before all right let's see um travis it looks like you know what one is go ahead and take the mic tell me what a rational function is not mistaken it's a it's a function that only has real numbers this is probably a prerequisite for any function but no i'm going off memory yet not entirely sure well you're partially correct i i love where you're starting because you're thinking of rational versus irrational numbers i love that j do you have anything to add to that go ahead and take the mic if you have anything that you can add um is it when like you divide uh i believe it said polynomials right there we go so if we combine those two definitions that travis and j gave us that's going to give us exactly what we're looking for whoops there we go so what is a rational function it's basically a function that is the ratio of two polynomials what's another word for ratio what does that look like type it in the chapter right it's a fraction yeah yep yep exactly a quotient yep right fraction good excellent guys all right so this is the way your textbook had it which looks really complicated and maybe even you might look at it and go oh gosh it seems so complicated but really it's just two fractions all right and with one polynomial over another one so this is the polynomial p of x and remember these are just variables it could be f of x it could be g of x anything like that right it's just any type of polynomial is really all that means one polynomial over another one there is a little qualification it's telling us that q of x or whatever is in your denominator um can't equal zero so hmm so let's think about that why um why do you think type it in the chat why can't why does q of x have to or can't equal to zero right andrew because you can't divide by zero right right what's that call what kind of fraction is that if you have a zero in the denominator and you remember that vocabulary word oh yes you guys do yeah nice yeah it's called undefined you got it andrea yeah so we don't want to have an undefined fraction so that's why it's telling you that it can't equal zero all right so that function at the bottom can't equal 0. all right so um let's see raise your hand if you think you could create a rational function just make one up andre i see your hand up i'm not sure if it was up from earlier can you create a rational function for us just make one up oh his hand went down oh but lucas's popped up lucas could you create one on the board with your um with your writing yeah yeah i can i can okay great all right just make one up it doesn't really matter what it looks like maybe any kind of rational function great numerator loving it sweet and don't forget your function notation so it's got to equal something all right great so we want to make it like um make it like f of x equals something yeah and x cannot equal zero perfect yeah yeah so go ahead and put f of x equals and you've got it okay we always want to remember to do that function notation because lucas did all the hard work over here right and he definitely doesn't if i was grading it or miss hunter was grading it we definitely won't don't want to count off because he forgot something like that right those are the most frustrating errors you're like oh no i forgot something so easy so excellent example love it love it that is a perfect example okay pretty good all right so let's keep moving all right so let's talk about here we go all right so we're going to talk about um vertical asymptotes and domains and they are actually related to each other all right so your vertical asymptotes and domain they they they have an effect on each other all right so to find your domain and the vertical asymptotes you're simply going to set your denominator equal to 0 and solve all right that's all you're going to do you're just going to be looking at that denominator setting it equal to 0 finding your solutions down there and that's what's going to help you find your domain and your vertical asymptote however for your vertical asymptote so a lot of times i'm going to use va or h a instead of typing out and you guys can do the same in your work you can type va for a vertical asymptote h a for horizontal asymptote if you're finding the horizontal asymptote although you're using the same process we have to remember to cancel out any common factors all right so if you had an x in the numerator in the denominator that you could cancel out or an x minus 2 you want to cancel those out first before you find those vertical asymptotes um these factors that you can cross out are called discontinuities okay in other words there are holes in your graph so you want to make sure and include those but that's when you're finding the vertical asymptote and we'll do some examples to show that but really all you're doing is working with the denominator for the most part to figure out what your domain and your vertical asymptotes are going to be all right so let's do an example all right so i know it's it's kind of easier at least for me to learn if i'm seeing an example besides just words so we kind of put it all together all right so this one's asking us to state the domain of the rational functions and the vertical asymptotes we're going to do both and i'm going to use the set because your multiple choice questions seem to prefer this set for the most part so we'll use these as our notations all right so we're going to have the domain and remember it's a set of it's x so it's x is a set of all real numbers and then we're going to find the exceptions and then we're going to find the vertical asymptote oh we haven't even done it yet logan [Laughter] all right let's try it first and then see if you understand a little better how about that all right so to find the domain and the vertical asymptote we're going to be looking at the denominator right so we're going to take that denominator and we're going to set it equal to 0. so let me start drawing so all i'm taking is just the denominator which is x to the third minus two x squared minus three x and i'm just gonna set it equal to zero and solve all right so when i do this first i want to look for a greatest common factor all right so let's look at the three terms we have three terms here right we have x to the third we have negative 2x squared and we have negative 3x so i see it in the chat what do you guys think what is that common factor i'm going to take out of all three right zachary we're going to take that x out perfect so let's take it out and see what's left over if we had x to the third but we took one x out how many x's have we got left two of them so we have x squared yeah well x squared all right and then if we took an x out of negative 2 x squared what's left we divided it out yep negative 2 x we have not changed the sign and then this last term we had a negative 3x we're going to take out that x right this is going to be just negative 3. perfect all right so we've taken that x out now we want to factor what's in parentheses because that's going to make it a lot easier to factor don't you guys agree all right so let's set up our factors and this is a nice factor so we need two numbers that multiply together that equal negative three but if you add those same two numbers together it equals negative two so what are those going to be all right so lauren says plus one and negative three so definitely i believe that's going to work yeah because negative let's see so we know they're going to be one and three right okay we just have to sometimes it's just kind of tricky which one is going to be negative and which one's going to be positive all right but you're exactly right because if you take positive 1 times negative 3 you get negative 3 and if you take positive 1 and add to it negative 3 you get negative 2. so it checks out perfect congratulations to all of you guys who got that correct all right so we've got all of our factors all right so i'm just working with the domain all right so this is the domain and then to find those restrictions on my domain i set all of those factors equal to zero remember x is indeed a factor oops what am i doing here let's take that away i'm just making up things as i go along which are not correct there we go how's that look a little better all right very good jacob you got it all right so we're just gonna solve for x now this one already is telling us x is equal to zero this one x is going to equal the negative 1 right because you're subtracting 1 from both sides this one we're going to add 3 to both sides that gives us 3 all right so we've got 0 negative 1 and 3 and remember our degree due to the fundamental theorem of algebra tells us how many solutions we're going to have and how many solutions do we have 0 negative 1 and 3. okay all right is this making a little bit more sense so all we did to find the domain of a rational function so a rational function remember it's a fraction okay you're going to take that denominator set it equal to 0 find your solutions okay logan we're going to get into that we're going to get into that so hang on for that okay because remember you may wonder okay well we'll just keep going okay so we're gonna talk about the vertical asymptote again with the vertical asymptote we have to determine if there are any common factors cancel out right so we had that 3x i'm just going to rewrite it with our factored denominator so if we had 3x i'll write it with a different color so we had 3x so in the numerator and then i'm just going to do our factored denominator okay so i didn't i'm not changing the problem i'm just doing what we factored in there okay okay now remember this is you cannot equal so these are they cannot equal zero it cannot equal negative one it cannot equal three and that's for your domain okay so it's a set of all real numbers except for zero negative one and three now for our vertical asymptote we have to see if there's any holes in our graph any discontinuities remember that's when you have a common factor in your numerator and your denominator so do we have that when we go to find our vertical asymptote is there anything we can cross out from the numerator and the denominator and this might be difficult to see yep savannah you are right olivia you are right yes we can cancel out the x and you may think how do you exactly see that mrs haney so remember 3 and x is the same as 3 times x if you want to you could take the x out of the three all right kind of like how we got the greatest common factor down here when we did this step right here you could take the the x out of the three because it's still x times 3 which is 3x right and then we still have our denominator right and so this might make it look a little clearer if you want to do that you don't have to but sometimes that might help so you can see how and remember this is all one term this is all one term down here so you can you can cancel these out so when you find your vertical asymptote get back to red it's going to be the same process but you have to take those discontinuities out those holes all right so now i'm still going to be looking at my denominator but the only thing i have left is the x plus 1 and the x minus 3. so those are the only ones i'm going to set equal to 0 to solve okay so ends the same as over here i'm subtracting one all right so my vertical asymptotes i have two of them i have one at x equals negative one and i have another one at x equals three now remember you can always always always graph this you can always graph it and just see plainly where those vertical asymptotes are sometimes you won't see the restrictions as clear but that should still stand out to you and you should be able to see it but again finding the domain and the vertical asymptote they are related to each other you're just going to be looking at the denominator however with the vertical asymptote you have to make sure you don't have any of those discontinuities okay and we're going to do another example like this too because you have one in your homework so when i do the ones like your homework we'll be doing another one like this all right and i do have this all nicely typed out for you because my handwriting's a little bit messy but not too bad and i went ahead and graphed it for you too okay so remember i told you you could graph it and verify it so remember i told you that we couldn't use that hole right that x equals zero so if you graph it do we have a vertical asymptote at x equals zero we don't yeah we don't no but we still have it over here at negative one which we knew we had one there right and x equals three so if you can see that one i put it in over here the dotted lines so i just graphed it to verify it and that's always helpful i think to graph it especially in module one i work harder if you don't have to okay and then remember i put about the discontinuity the whole must be excluded from the domain therefore there's a hole at x equals zero yeah it's not too bad yeah and we're going to do another example of it yeah just graph it to verify you've got it right it's not going to show up j it's not going to show up it's not going to show up on your graph you have to determine it because we had that x that we cancelled out there from the numerator and the denominator yeah graphing will not show your holes all right so let's talk about horizontal asymptote and range so we just talked about domain and vertical asymptotes and how they're related so the horizontal asymptote and the range they're also related all right so let's take a look at these so to find a horizontal asymptote what you're going to do is you're just going to be looking at the degree of the numerator and the degree of the denominator all right so again what's a degree and then remember raise your hand if you remember what a degree is what's the degree emma go ahead take the mic what's the degree um degree the degree is always the highest leading coefficient close it's near the the leading coefficient but it's not actually the coefficient itself it's the what emma oh the exponent yeah there you go you got it i know i'm sorry that's not right i knew what you meant very good nicely done emma i knew she had it all right so what we're doing is we're just looking at the largest exponent in the numerator and the denominator and that's going to tell us what our horizontal asymptote is without even graphing so weird stuff all around all right so let's look at all three different scenarios there's three possibilities so the first possibility which it doesn't really matter which one's first really is the degree of the numerator is less than the degree of the denominator so take a look at here our degree in the numerator is 2 and the denominator is 3. so if the degree of that numerator is less than the degree of the denominator your horizontal asymptote remember h a horizontal asymptote is y equals zero and it doesn't matter how much greater that denominator degree is then the top it could be a 1 up here and it could be a 2000 as your exponent down there it doesn't matter any time the degree of the numerator is less than the degree of the denominator your horizontal asymptote is always at y equals zero okay and the leading coefficients they don't matter either okay just always look at those degrees every single time y equals zero you could graph any kind of rational function you're gonna find that out okay so make sure you have that in your notes take a screenshot of this one all right let's look at our second scenario second scenario is when they're exactly the same those degrees so let's look at our example so here they both have a degree of three all right so when that happens you are going to take a look at your leading coefficients and make a fraction out of them so it's going to be the leading coefficient of your numerator over the leading coefficient of your denominator so what is the leading coefficient of this numerator there's no number up there but what is it yeah right it's a one let's go ahead and just write it in just so it makes a little bit more sense okay so when those degrees are the same you just look at your leading coefficients right here and you make a fraction out of it that is your horizontal asymptote remember horizontal asymptotes are always going to be a y equals just as your vertical asymptotes will be an x equal all right so that's our second scenario and then we have one more scenario to look at four horizontal asymptotes and that's when the degree of the numerator is greater than the degree of the denominator so let's take a look at our examples so we have x to the fourth over three x to the third so our degree of our numerator is four and the denominator is three so that one is definitely greater than that one then there is no horizontal asymptote and it doesn't matter how much greater that numerator degree is than your denominator degree it will always be no asymptote okay no horizontal asymptote and don't get confused with equals zero and not having one because that's a big difference i've seen that before in the past where students would say it equals zero but there is a line on the graph where y equals zero right that's right along our x-axis so that exists so if there's no horizontal asymptote it doesn't don't get mixed up between the y equals zero and no horizontal asymptotes okay all right and one more thing i just wanted to add because it is going to be on your assessment is a little note when the degree of the numerator is exactly one more than the denominator then you have a slant asymptote okay so this one here would this one have a slant asymptote is it exactly one more yes exactly so this one there's no horizontal asymptote because it but there's going to be a slant one it has to be exactly one now for no horizontal asymptote it can be a 25 up here and a three down here and then there wouldn't be a slant okay so it's only going to be one more um hannah great uh question if there were not any variables up here like if you just had the number five then that would be x to the zero x because there were no zeros right i like i like your your logic there good point all right let's keep moving let's do an example of it huh it's always a little easier to understand when you take a look at an example all right so this is our horizontal asymptotes and our ranges again always graph these things it's kind of things in perspective but also going to verify if you're on the right track or not all right so let's take a look at our range and remember the range and the horizontal asymptote they um have something in common all right so let's figure out the horizontal asymptote first okay so in order to do that we want to look at our exponents okay our degrees our highest exponents so are they the is one greater than the other is the numerator greater than the denominator denominator greater than the numerator or are they exactly the same right right yeah you're right alexandra they're exactly the same so when they're the same how do i determine what is that horizontal asymptote do you remember what we do right we're going to make a fraction good john yeah we're making it a fraction we sure are jonah right so you're going to take the leading coefficient over the leading coefficient and remember there's not a leading coefficient here but we know it's 1. i'll go ahead and write that in just so that we know okay so we're going to take that leading coefficient over the leading coefficient when the degrees are the same which they are and that's going to be our horizontal asymptote which is 3 over 1 right oh that looks like a question mark that should be a 3. so it's 3. and don't forget the y equals part because remember it's a line so it's actually here let's see here it is right here and if you graph it you're going to say well my answer it makes sense so so that works out good all right now we're going to look at our range okay and to find the range i always recommend definitely definitely definitely to graph it it's going to be the easiest way to figure out your range and remember the range is what what can equal so your x or your y it's your y yeah okay so remember i said that this plays a role so if there is a horizontal asymptote at y equals three which there is you're not going to have a value of the range there are you but it looks like you're going to have every other value huh right so i'm going to go ahead and create my range which would be i'm going to start with negative infinity remember i'm looking at my y values so down here i've got negative infinity don't i and it goes up up up to three oops let me get my eraser out a little crazy here with this let's take that out okay there we go all right so negative infinity to 3 and then again i'm gonna have another set and what's going to be in my next set so i've got negative infinity to 3 on the y value what's going to be in my next set of parentheses this part here right there you go yeah good hannah yeah you got it matthew so it's gonna be three up to infinity so as you can see graphing is going to make this one so much easier it's going to make it so much easier all right so when you have a question like this find your horizontal asymptote first okay so you can just do depending on your degrees find your horizontal asymptote graph it make sure you've found your horizontal asymptote correctly and then figure out what your range is all right yes that's a union yeah so that means that your range is from negative infinity to three and three to infinity so it kind of works here excellent all right and there's my answers nicely typed out it's much prettier than my writing yes madison if the degrees are the same you just do leading coefficient over leading coefficient yes now if the numerator the degree would have been larger then there would not have been any horizontal asymptote at all if it would have been smaller then it would have been y equals zero yep i can go back a slide real quick talking about that one all right you're welcome all right we're going to keep moving all right so just remember some key points the vertical asymptotes they have an effect on the domain and your horizontal asymptote will have an effect on the range okay so just keep those in mind all right so i've got question time so i'm going to pause the recording briefly there we go alrighty all right so i wanted to show you some examples that you're going to see on your mid module exam yes not only do you have an exam at the end of the module you also have one right in the middle all right and this one a lot of students have asked about the this is combining both the horizontal asymptote and the vertical asymptote so this is an example of what you might see on your exams so it says which of the following functions rational functions and then they've given us a specific horizontal asymptote and a specific vertical asymptote all right so let's see what we can find here and see what we discover okay all right let's do it all right so the easiest thing to start with really is going to be the horizontal asymptotes and we just did that it's fresh on our mind okay so we're going to have to work backwards compared to what we did i love how you're doing the process of elimination i'm going to do the same same skill all right if our horizontal asymptote is at a number it's at y equals 5 what do we know about our degrees think about that give yourself a moment and think about it oh you guys are on this yeah your degrees have to be equal all right so they have to be exactly the same perfect all right so they have to be the same so let's just check all of our answers x squared x squared x squared x squared uh that doesn't let us eliminate anything does it okay so let's go again if it has to be at y equals five remember it has to be leading coefficient over leading coefficient so the fraction is going to look like 5 over 1 it might be 10 over 2 something like that so let's see which ones are going to work would a work yeah a would work remember there's a one there let's just go ahead and put all these ones in just so you see them so that one would work would this one work b no no that's not going to work nope because that would be one right that would be horizontal absorption for one so that's not going to work how about that one will it work yep it'll work what about this last one nope let's get rid of it we hate it it's gone get out of here d yeah we don't want you all right so that has eliminated uh b and c um lydia that they don't work a and c they do work so they do work because if you take the leading coefficient over the leading coefficient 5 over 1 that equals 5 and that's what we're looking for remember that's going to be 5 over 1. this one here is also 5 over 1. yeah all right and next we've got to find the vertical asymptotes at x equals 2 and x equals negative 4. now we did this before but we're going to have to go backwards this time aren't we so what we're going to have to do is we're going to take our solutions and go backwards so what's the first step we're going to do with those solutions what are we going to do what do we have to create right we have to create some factors right so if our solutions are x equals 2 and x equals negative 4 what would those two factors be and we're going to have the opposite sign there you go you got it x equals 2 and x equals positive 4 or x plus 4 sorry that x equals x minus 2 and x plus 4. you guys got it don't listen to me look in the chat you guys are on it yeah so i just took those solutions and i made factors out of them because i'm going to have to expand them am i not so you guys go ahead and expand it go ahead and do your multiplication is it going to be x squared plus 2x minus 8 or is it going to be x squared minus 2x plus 8. do that work and let me know what do you find out getting a lot of a's yep you're gonna have to foil it and i totally agree yeah if you expand that out it's gonna be x squared plus two x minus eight which would make a your answer so we took two skills together to do this one all right so just as a brief brief um recap it's always easiest to do your horizontal asymptote first you are some smarty pants love it find out your horizontal asymptote get rid of those answers that don't make sense right so b and d we knew that couldn't be a possibility because that horizontal asymptote was not at y equals 5. always graph these individually if you wanted it the long way and then next we determined with our vertical asymptotes remember that's always the factors or the solutions of the denominator so if we had the solutions we had to go back instead of factoring and then solving we went backwards all right so it would be x minus 2 and x plus 4 remember it's always the opposite sign and then we foiled we distributed whatever you want to call it and we got a nice work guys nice work you're going to do great on that question on your mid-module exam love it all right and this is a question like you're going to see on your actual exam all right boy it's going to be tough we got all kinds of clothing items in the chat all right so let's take a look at this one so we want to know what is the equation of the rational function with its corresponding slant asymptote so this one's not bad but we're going to do a little bit of work here all right so first off to have a slant asymptote just as a reminder the degree of the numerator has to be one more than the degree of the denominator so let's see if we can eliminate any of these a b c or d let's see this one that's two and that's one that doesn't that's that could work two and one two and one oh we can't eliminate anything there so darn we tried all right so the next thing is is you want to determine which one that is a graph of okay because they're going to give you the graph and you have to determine so i always think it's really easy to find out where it crosses the y intercept okay so whenever you so we want to see it's a little over five you can't really tell here but it's five point something so and you don't have to do it this way you can do it many many many different ways this is just the way i'm doing it because i'm lazy i like to do an easy way so that would mean that x has to equal 0. so x if x equals 0 i want to see where it crosses the y axis so i'm just going to take and there's really only two functions to choose from either this one or this one with the plus because the numerators are the same in each one so if i made that that x0 so that would be zero squared minus sixteen over zero minus three i get that it's going to be positive sixteen over three which is 5 point something so that is a positive value so that one could work it might work um but if my denominator was x plus 3 basically you're getting rid of this and this that's gonna be a negative five point something right yeah so i know that that graph is not going to match so i'm gonna eliminate this one and i'm going to eliminate this one remember you don't have to you can graph these um and then the last part is that we have to decide if this is our correct function we have to decide which one is the correct slant asymptote you could also graph that and see which one makes sense you can see if it's x minus 3 or x plus 3 or you can do your division and that's what i'm going to teach up here all right so if we do our division all right i want to see which one gives me a slant asymptote so i would take my numerator which is x squared minus 16 and i'm going to divide it by my denominator so you can do long division but i'm going to do synthetic division the only thing that's tricky about synthetic division is you always have to have a placeholder and what i mean by that is we have an x squared but we don't have an x to the first power all right so i'm just going to write it with an 0 x to the first power minus 16. this is exactly the same as this but i need that placeholder all right i just need it for my synthetic division right so then i'm going to take my coefficients so my coefficients are 1 0 and negative 16. and then i'm going to divide it by my denominator okay my denominator is x minus 3 here okay for either one of those if you set it equal to 0 and solve for x that's what you're going to divide by all right so if you remember synthetic division you drop down the first one so the one goes down then you multiply three times one is three add the two together three plus zero gives us three and you can actually stop here on this one but yep drop mama exactly and then you can do three times three which is nine and this is actually your remainder here so you don't have to do this math if you want to we really just want these two here and those are the numbers we're going to use to determine sorry looking for the mouse to determine what our slant asymptote is all right okay and then when you figure it out when you put it back in you're using those two these are going to be your coefficients you're going to have your degree is going to be one less so if we started with x squared it's going to be x to the first so it's just going to be 1x plus 3 which is basically x plus 3. this is just synthetic division and that is your correct answer right here and remember you can always graph it if this confuses you christina just graph it if you graphed it you would see that you had a uh it would be here you had a slant asymptote that went right in here like that yep i will do more of this uh this division absolutely yes you're welcome all right so it's time for extra practice and these are going to be questions like your assessment so i'm going to go ahead and stop the recording however you are welcome to stay and ask any questions i've got examples from each one that are on your assessment