Stony Brook MAT 123 Fall 2015
Lecture 14: Rational functions
October 12, 2015

Start   like i said if you are not yet signed up for the part 1 retake, you need to sign up you should have an email from professor sutherland you should sign up if youre taking it in the testing center this week if you dont take it this week you dont take it al all
0:30and you are stuck with your bad grade the midterm is 2 weeks from tomorrow the midterm will consist of were not quite sure but certainly the stuff we did the last few days cognitive division stuff with circles and ellipses and then logarithms which is coming i know youre love logarithms so keep in much right now were are gonna do a little on rational functions, were not gonna do to much
1:01so a rational function is a function/function, ratio so basically the function became a fraction so if you have f(x)/g(x) that is a rational function and of course at out of all these things the bottom cant be zero so thats always very important if the bottom is 0 you would have a problem
1:31has everyone gotten some?
pass around are you guys all coming to homecoming on saturday?
its a big game you dont have tickets?
oh you just go to the athletic center and get tickets in the booth there i dont know hope a lot do you guys know if theres plenty extra tickets well you guys can get some, i already got mine
2:00but i have season tickets there you go i get took at the back of his head for most of the game a lot of too you can see me i right by the 45 yard line about five rows up near all the other old folks right there in the center youre suppose to say youre not old yea its a little late for that so a rational function
2:31is a function over a function you only need a couple of things to worry about suppose its something that looks like this alright so x-3/x+2 what do we know about this function?
we have a problem right there at x-2 at x =-2 we have 0 in the denominator, which the function is not going to define there
3:01so at x=-2 you will get a vertical asymptote or more general, what if the bottom is 0?
and the top is not also 0 where the bottom is 0 and the top is now also 0 you will get a vertical asymptope so a vertical asymptote means the function will go up infinity or down minus infinity so if you were graphing it you know what graphs look like
3:30you can have more than one of these of course the second possibility, the second thing you have to watch out for are horizontal asymptotes as known as end behavior, so im gonna cover this up so rational functions use a graph they often look something like this youll get something that looks like that, see it has vertical asymptotes and horizontal asymptotes obviously there's many variations of this
4:02the verticle asymptotes are where the denominator is zero horizontal asymptotes has to do with the ratios of the two functions say we had something that looked like that were not gonna ask you to graph it, you can do that in a calculator and this will have a vertical asymptote
4:322 places it has vertical asymptotes where the bottom is 0. so thats 1.. and x equals -1 plus or minus 1 nice and simple, so if you are graphing it for example, this is not this but if you were graphing it [sneeze]
5:00you are very blessed have one at -1 the curve is either gonna wind up or wind down, it depends on the signs so you sorta of have to find out where this is positive and where this is negative and the functions also has a horizontal asymptote.
the horizontal asymptote is the end behavoir it means what happens when i say very large and there you have to look at how fast the top of the functio is growing and how fast the bottom of the function is growing
5:30so if you put in a number like 20.. the top is 55 and the bottom is 21 times 19 so thats a much bigger number alright because 99 so at a thousand the bottom is starting to get much bigger then he top and a million the top is so much smaller then the bottom that it starts to look like 0 so this would have a horizontal asymptote at 0 and thats what they call the end behavior
6:02this graph has to go out along the x axis you know to do something like that or come up like this id have to think about the graph if thats all we know about a horizontal asymptote, there are some rules as far as our lost post, help you figure end behavior
6:37so you have a polynomial on top and a polynomial on the bottom so the ax to the m will be the highest term power on the top even if theyre not written in order its whatever term on top had the higher power similar to the bottom, bx to the n whatever term on the bottom has the highest power okay we will do actually examples in a moment
7:01then theres 3 possibilities, either the top is bigger then the bottom the bottom bigger than the top or there equal these are not complicated right?
bigger. smaller. or the same if the top is bigger then the bottom there is no horizontal asymptote if the top power is bigger than the bottom power, what that means is
7:30as the numbers start to get big the top will be so much bigger than the bottom it will dominate the function and the function will go through infinity we will do an actually example in a sec if the bottom is bigger then the top the horizontal asymptote is y=0, also known as the x-axis so it will go through and slice the x-axis
8:03and if theyre equal the horizontal asymptote is a/b hope you guys are following for actual math content
8:46okay lets do some example, do we have this written down?
9:42so if we had?
so we have something like this
10:10well it says the second but ill make it the 3rd it doesnt matter okay, the only thing that matters is it doesnt matter about the cubed or squared root it only matters the highest power of each one because when you start to get to a really big number, like a million a million to fourth is a lot bigger than a million to the third you guys who have taken chemistry you know your scientific notation
10:31a million to the 4th is 10 to the 24th a million to the 3rd is only 10 to the 8th big difference, theres a million concepts similarly you only care about the highest term on the bottom so you only care about these two terms now since the term on top is bigger than the term on the bottom no horizontal asymptote
11:02we wouldnt ask you to graph this, its messy what if it said now i have the same power on the top and bottom so even though this is cubed and this has an x, it doesnt matter
11:31only thing that matters is the highest powers all the terms start to not matter when its very large you can use your calculator to see if you want so this will look like, 3x to the 4th over 5 x to the 4th so the horizontal asymptote will be 305
12:10alright and one other possibility now i have 5x to the 5th on the bottom for those of you who cant read my hand writting
12:33so again it doesnt matter whats going on on top if the bottom power is bigger than the top power so now the horizontal asymptote is the x-axis y=0 that make sense?
were actually gonna learn how to draw one of these
13:01get excited very excited so far so good? alright im going to cover this up no you guys are going slowly over ther
i cant wait any longer im not getting any youger lets do one of these
14:10suppose i want to graph that see you very nervous and youre like ugh seriously graphing
14:31look ill need very good doctors when the time comes i want to know where my a students are dont want any of the b students doing any surgeries just a students alright so we can figure out a few things right away from looking at it first of all we notice the vertical asymptotes vertical asymptotes where the bottom is 0
15:47bigger power on the bottom the horizontal asymptote is 0 this curve eventually go out to the x axis what else do i know?
when x=2, what does this equal to?
16:01when x equals 2 the top is 0 bottom is not 0 any fraction with a zero on top and not on the bottom is 0 okay so im gonna cross the x axis and have an xintercept x=2 so again if you plug in 2 on top you get 0 you plus in 2 on the bottom and you get something else and dont really care 0 over numbers equals 0 have you realized that?
16:31this is going to go right here gonna go from 2 other what happens when i plug in 0 thats the y intercept right?
well lets see when i plug in 0 i get negative 2 over 3 times 4 over negative 12 so negative 2 over negative 12 is 1/6 hat would be this spot
17:06so now i have a pretty good idea what this graph looks like, believe it or not in the middle this graph has to look like that, how do i know that?
well lets see it has to go through 1/6 has to go through 2 these are vertical asymptotes so when it gets close to the asymptote you have to go to infinity how do i know it goes up here rather down
17:30because if it went down i would have another x intercept so if it went f=down through here it would cut through the x axis and it doesnt cut through the x axis it has to go up here i guess it could bounce and go up and just sort of touch the x axis and go up its possible pick a number through 3 and 4 and see what happens you plug in 3 you get 1 on top you 6 times -1, you get a negative on the bottom you get a negative number so you must be going down
18:01do it again just sign test it, you only care if its positive or negative take a number between 2 and 4 and thats 3 and you plug it in each part you plug in 3 here and you get a positive number plug in 3 here you get a positive number, and plug in 3 here you get a negative number positive, positive, negative will make the whole thing negative you guys get that? no?
do it again when you plug in 3, this is 3-2
18:30this is 3+3 and this is 3+4 thats equal to 1 6 times negative 1, thats negative you dont care what the actual value is in the sketching so thats how i know it goes down here rather than up so dont worry i dont expect you guys to be able to do this instantly, just follow along now what happens when x is bigger then 5 on either, i got it here on the asymptote and later i have to get this asymptote so i either i either go like this or go like that and i got to figure out which one
19:03if you just take any number bigger than 4 like 5 and you plug it in and see if you get a positive or negative if you get a positive it must be this and if i get a negative it must be that that make sense? Yes?
the y intercept is when y equals 0, you plug in 0 and get negative 2 3 -4 you multiply it out you get 1/6
19:32so what you are doing is you just want to find the signs of things so if i say x is 5 if i get a positive value ill be up here, if i get a negative value ill be down there and you know it has to have one of those two shapes it has to have this asymptote and this asymptote so you put in a number like 5 5 minus 2 on top is positive 5 plus 3 is positive 5 minus 4 is positive, the whole this is positive so
20:02its gotta be that well practice more than one of these and now i take a number to the left of negative 3 to figure out if its doing this or doing that so i pick a number like -4 i go to the top -4-2 thats -6, thats negative -4+3 thats -1 so thats negative -4-4, tis negative 8 thats negative
20:30so negative negative negative 3 negatives means negatives right?
because 2 negatives is positive so one negative makes it negative again and youre done, you dont have to be anymore precise than tha okay so what did we do.... we figured it out where it crosses the axis we figured out the asymptotes and then we just figure out if its positive or negative you all long a bit stunned so well do alittle more of these
21:05can slightly use the other one this will be nice and simple like that simple for me now you guys are doing it for
21:40so were gonna find some key things first well find where the vertical asymptote is the vertical asymptote is where the bottom is equal to 0 so wheres the bottom equal to 0 x equals -2
22:10then i look and say okay how about a horizontal asymptote, well the power of the top is x the power of the bottom is x they are the same but look at the coefficents 1/1 so its just y=1 this is x to the 1 on top and this is x to the one on the bottom so same pwero
22:30we just care what the coefficient of this is and that is. its y=1 so that tells me when they go out to infinity, im gonna be at1 y=1 again, sure okay so the power of this is x to the 1 and the power of this is x to the 1 remember the rule, when the powers are the same its just whatever that coefficient is, a.
over this coefficient b. well theyre both 1 right? 1x to the 1x
23:01so 1/1 is 1 okay, you sure?
so now we know when i go to infinity im getting to 1, so the question is just whether im gonna go above or below to get to 1 well lets see whats the x intercept another when the top is 0 top is 0 and x=4 when i get x=4, i get 0 on top and 6 on the bottom 0/6 is 0 so when x=4
23:32i go through the x axis now with the y intercept, with the y intercept i plug in x=0 so i get 0 minus 4 on top 0+2 on the bottom so minus 4/2 is -2
24:00now im gonna get y=-2 which is.. that looks about right now we just have to sorta figure out what happens okay i have avertical asymptote and i have a horizontal asymptote and i have to go from -2,4 so i probably do something like thae the only place i cross the x axis is here the only place i cross the y axis is here and i have to get down there some how and there somehow so on the other side im either going to have this
24:32or im gonna have this how do i know?
well you have two possibilities, one you can take the number to the left of -2, like -3 and see if you are above 1 or below 1 but i also know something if the bottom curve is right there, were gonna have an x intercept somewhere around there i dont have anymore x intercepts, i only have the one' so its gotta be the top arch thats it thats the whole graph
25:04how we feeling about these?
want to practice some more, lets practice some okay any you guys want to do nice simple one
25:41how bout that, lets see if you can graph that so find the vertical asymptote, the horizontal asymptote, the x's, the y intercept and sketch the graph
28:07okay to find the x intercept you set the numerator equal to 0 to find the y intercept you let x=0
30:11do we need onemore minute or are we ready
30:52alright thats probably long enough
31:17okay so va stands for vertical asymptote whats the vertical asymptote, when you take the bottom and set it equal to 0 you get x=-1
31:30thats where the bottom is 0 thats easy horizontal asymptote well now we look at the power of the top which is 3x and the power of the bottom is x so we just do the ratio get 3 and 1, so its 3/1 is 3 and thats a 3 so far so good?
so the power of the top is 3x so the highest power is 3x
32:02the highest term, the highest term on the bottom is x so we look at the ratio, 3/1 include 1x so 3/1 is 3 we have a horizontal asymptote of 3 we have a vertical asympote at -1 okay lets find some intercepts the x intercept is when the top is equal to to 0. when the numerator is 0 which is x=, sorry
32:32x=5/3 cause again when x=5/3 the top is 0 0 over anything other than 0 is 0.
okay that means as long as you have 0 the whole thing is 0. okay? thats the x intercept the y intercept if you plug in 0 you plug in 0 on top you get -5 you plug in 0 on the bottom you get 1
33:01the y intercept, is y=-5 itd be down there that means the graph up. and go something like that on the right side okay how we doing so far?
happy love life you only learn so much in math class now we can figure out the left side well.. the left side cant look like this
33:31because if it looks like that we have an x intercept here so it must look like that the other way you can test that is take a number to the left of -1, like 2 and we plug in -2 here you get a number above the horizontal asymptote, a number bigger than 3 than its this branch, if you get a number less than 3 you get this branch that make sense?
34:00how we do on this one good yes howd i get which one? negative 5?
when i plug in 0 on top i get -5 when i plugin 0 on the bottom i get 1 so -5/1 is -5 oh right good so nick raised a good point the previous graph or couple graphs ago i crossed the horizontal asymptotes you can cross horizontal asymptotes but you cannot cross vertical asymptotes
34:32a horizontal asymptote is really just a description on what is going on outside of infinity but you can cross, you can cross as many times as you want if i plug in the number -2 it will be above 3 the other way you know is so theres two ways to tell what the left side looks like one is pick a number to the left of -2, -1
35:02see what happens the other is just remember if the number to the left of -1 where the graph looks like this you have to cross the x axis which means you have another x intercept which you dont, you only have 1 intercept lets do another 1 slightly more dificult
35:32-5 once you plug in x=0 you plug in x=0 on top and get -5 you plug in 0 on the bottom you get 1 -5/1 is -5 so plug in -2 you get a=-6-5 over -2+1 -11/1=11
36:00so when x is -2, y is 11 which is above 3, so that means you must be above the curve not below the curve alright lets do one thats slightly more difficult i will work you guys up to a hard one and then we will be very impressed
36:43how bout that?
so a couple of rules for everybody to remember
37:10alright if you want to find the vertical asymptote you set the denominator equal to 0 you want to find the x intercepts you set the numerator equal to 0
37:38so to find the y intercept we plug in 0 for x okay to find the horizontal asymptotes you need our rule
38:05you need the rule i gave you before and then to figure anything else out you do whats called sine test figure out where its positive and where its negative and thats enough to come up with a basic sketch of the graph doesnt have to be perfect you just have to know those things can you guys see that on the right side of the room?
so lets figure this out, lets find
38:33vertical asymptote set the bottom equal to 0, its equal to 0 in two places. x=4 and x=-4 find the horizontal asymptotes the highest power on top
39:01is just 2x, just 1 and the highest power on the bottom well its gonna be x squared multiplied out so youll get an x on top and a x squared on the bottom so thats gonna be y=0 or the x axis did we find the asymptotes okay?
so far so good?
yes what i have 2 possible vertical asymptotes the bottom is 0 in the denominator is 0 at 4 the denominator is 0 at negative 4
39:30there are two places where this gonna be asymptotes, yes?
the top is greater than the bottom if the top is less than the bottom im not really gonna do to many of those alright now i want to find the x intercept, so i set the top equal to 0
40:00and thats what 2x+1=0 or x is -1/2 and to find the y intercept to find the x intercept, take 2x+1 set it equal to 0 and get x=-1/2 find the y intercept set x=0 and get 1 on top -4 and 4 on the bottom which is 16
40:31so y=-1/16 okay lets see i have -1/2 -1/16 so my guess is this looks like thay how you doing so far?
doing good?
now lets try and figure out what the left side of -4 looks like
41:00you pick a number to the left of -4 like -5 plug it in the top and you get -11 -9 you plug it in the bottom you get -9 times -1 so a negative, negative, negative so 3 negatives equal a negative the graph goes down like that you got to pick a number to the right of 4 like 5 the top would be positive, this will be positive, this will be positive, so the whole thing will be positive
41:34look something like that anybody able to get the right graph?
some of you? yay!
not with this variation, depends on the minus signs and the plus signs notice we cross the horizontal asymptote right here thats okay alright you just cant cross the horizontal asymptote way out there now ill do one more to make sure everyone has the idea and ten we will do something else
42:05how we feeling about these?
i plug in 0 for x to get the y intercept so the top is 1 0 plus 1 -4 and positive 4 is -16 we feel good about this one?
well do one more, lets make it a little messier
42:30see how good you guys are i factored it for you still some shakiness thats good
43:01this is al the energy we are putting in for rational functions not more than this so first lets find the vertical asymptotes vertical asymptotes, where the bottom is equal to 0 its equal to 0 at -4 its equal to 0 at positive 3 so wehen graphing this we have a verticle asymptote at -4 we have a vertical asymptote at positive 3
43:32so far so good?
that was easy thats the fun part now i got to find horizontal asymptote okay you can have many vertical asymptotes you can have no more than 1 horizontal asymptote unless we give you a really bazar function but were not gonna do it okay?
so horizontal asymptotes, well if you multiply out the top you get x squared and something and you multiply out the bottom you get x squared something
44:00so x squared over x squared is 1 so the horizontal asymptote is is y=1, so agian x times x gives you x squared x times x gives you x squared so this is x squared and the bottom is x squared 1/1 its not 2x squared its not x cubed its just x squared and x squared okay x intercept
44:32theres 2 x intercept there are two places where the numorator is 0 at x=1 and x=-2 agin you can have multiple places where x intercepts you only have 1 y intercept you have multiple intercepts the graph goes through -2 and -1 y intercept
45:00you get by plugging in 0 so you plug in 0 on top what do you get, you get 0-1 times 0+2 the bottom you get 0+4 times 0-3 alright thats plugging in 0 for all the x's so thats -1 times 2 thats -2 -4 times 3 thats -12
45:30so thats 1/6 that looks like a good spot for 1/6 so far so good?
so this graph gotta go through there probably look something like that in the middle how do i know it doesnt go up and down?
you can try some values if you are not sure also ill give you a clue if this is gonna bounce one of these terms are gonna be squared so its gonna go like
46:03up like that, yes?
horizontal, oh yes i didnt draw that sorry thats the horizontal asymptote okay?
so far so good?
its hard to sketch these, its easy to find points but hard to sketch them alright now whats happened to the left of -4 are we above the x axis or below the x axis take anumber like -5
46:30this is negative this is negative negative this is negative 4 negatives, an even number of negatives can make something positive negative times negative times negative times negative so gonna look like that okay take a number to the right of 3 like 4 positive, 4-1 is positive positive, 4+2 is positive 4+4 is positive
47:004-3 is positive everything is positive so the whole thing is positive, so again it looks like that these are painful i dont think we will give you antything this hard we might not do graphs at all we might just say find the points okay we may ask you the end behavior im not sure questions, yes?
how do i know this goes down here?
how do i know it doesnt go up like this?
47:33well first of all i know i love graphing pick a number between 1 and 3 like 2 and see if you are above the x axis or below the x axis also take a number between -2 and -4 like -3 see if you go up or you go down and the clue is when you get this when you bounce off the axis like that one of these factors will be squared
48:00or to an even power thats whats causing, whats happening is were getting sign changes every time, change. every time you get between the 0's the sign of this is changing okay as the sign changes notice this is positive then its negative and positive and negative and positive see you figure out when its above the axis and when its below the axis
48:36wanna take a number left of -4 -5, -10, negative billion what you can do is make a sine chart if youre not sure one thing you can do you take the 0's of this and put them on the x axis
49:00not on an axis -4,2,1,3 and then you can try and number each of these zones and see whether this expression is positive or negative its gonna do that when you try numbers okay?
where do i get those numbers, i take a number less than -4, like -5 and i plug it in and find the whole this is positive i find a number between -4 and -2 like -3 and i plug it in and find it is negative
49:31and that will help you figure out when you go above and below the x axis this is called sin chart or sin testing sine testing is often a very good way to find out whats going on with a function
50:14how did we do on these?
okay now were gonna lead to exponential functions so exponential functions and logarithms are closely related so lets give a few minutes to do exponential functions heres what happens
50:30so far we have been doing x to a number now we are gonna do a number to the x okay thats an exponential function 3 to the x, 2 to the x, 5 to the x they exhibit a specific kind of growth the type you see in biology and chemistry biology you see a lot of exponential gorwth, for physics you see radio activity you see all sorts of things, you see the sound so when you ring a bell the sound dies off exponentially
51:01all sorts of things and in order to solve exponential equations we are gonna use our favorite thing from math logarithms okay if get stuck on them youll get an extra point on the tes so thats youre homework ,learn about logartihms wednesday were gonna do exponential functions