# Finding horizontal and vertical asymptotes | Rational expressions | Algebra II | Khan Academy

Voiceover: We have F of X

is equal to three X squared minus 18X minus 81, over

six X squared minus 54. Now what I want to do in this video is find the equations for the horizontal and vertical asymptotes and I encourage you to

pause the video right now and try to work it out on your own before I try to work through it. I’m assuming you’ve had a go at it. Let’s think about each of them. Let’s first think about

the horizontal asymptote, see if there at least is one. The horizontal asymptote

is really what is the line, the horizontal line that F of X approaches as the absolute value of X approaches, as the absolute value

of X approaches infinity or you could say what does F of X approach as X approaches infinity and what does F of X approach as X approaches negative infinity. There’s a couple of ways

you could think about it. Let me just rewrite the

definition of F of X right over here. It’s three X squared minus 18X minus 81. All of that over six X squared minus 54. Now there’s two ways you

could think about it. One you could say, okay, as X as the absolute value of X becomes larger and larger and larger, the highest degree terms in the numerator and the denominator are going to dominate. What are the highest degree terms? Well the numerator you

have three X squared and in the denominator

you have six X squared. As X approaches, as

the absolute value of X approaches infinity, these two terms are going to dominate. F of X is going to become

approximately three X squared over six X squared. These other terms are going to matter less obviously minus 54 isn’t

going to grow at all and minus 18X is going to grow much slower than the three X squared, the highest degree terms are

going to be what dominates. If we look at just those terms then you could think of

simplifying it in this way. F of X is going to get closer and closer to 3/6 or 1/2. You could say that there’s

a horizontal asymptote at Y is equal to 1/2. Another way we could

have thought about this if you don’t like this whole little bit of hand wavy argument that

these two terms dominate is that we can divide the

numerator and the denominator by the highest degree or X

raised to the highest power in the numerator and the denominator. The highest degree term is

X squared in the numerator. Let’s divide the numerator

and the denominator or I should say the highest degree term in the numerator and the

denominator is X squared. Let’s divide both the numerator and denominator by that. If you multiply the numerator

times one over X squared and the denominator

times one over X squared. Notice we’re not changing the value of the entire expression,

we’re just multiplying it times one if we assume

X is not equal zero. We get two. In our numerator, let’s

see three X squared divided by X squared is going to be three minus 18 over X minus 81 over X squared and then all of that over six X squared times one over X squared,

this is going to be six and then minus 54 over X squared. What’s going to happen? If you want to think in terms of if you want to think of limits as something approaches infinity. If you want to say the limit as X approaches infinity here. What’s going to happen? Well this, this and that

are going to approach zero so you’re going to approach 3/6 or 1/2. Now, if you say this X

approaches negative infinity, it would be the same thing. This, this and this approach zero and once again you approach 1/2. That’s the horizontal asymptote. Y is equal to 1/2. Let’s think about the vertical asymptotes. Let me write that down right over here. Let me scroll over a little bit. Vertical asymptote or possibly asymptotes. Vertical maybe there is more than one. Now it might be very tempting to say, “Okay, you hit a vertical asymptote” “whenever the denominator equals to zero” “which would make this

rational expression undefined” and as we’ll see for this case that is not exactly right. Just making the denominator

equal to zero by itself will not make a vertical asymptote. It will definitely be a place where the function is undefined but by itself it does not

make a vertical asymptote. Let’s just think about this

denominator right over here so we can factor it out. Actually let’s factor out the numerator and the denominator. We can rewrite this as F of

X is equal to the numerator is clearly every term

is divisible by three so let’s factor out three. It’s going to be three times X squared minus six X minus 27. All of that over the denominator each term is divisible by six. Six times X squared minus 9 and let’s see if we can

factor the numerators and denominators out further. This is going to be F of

X is equal to three times let’s see, two numbers,

their product is negative 27, their sum is negative six. Negative nine and three seem to work. You could have X minus

nine times X plus three. Just factor the numerator

over the denominator. This is the difference of

squares right over here. This would be X minus

three times X plus three. When does the denominator equal zero? The denominator equals zero when X is equal to positive three or X is equal to negative three. Now I encourage you to pause

this video for a second. Think about are both of

these vertical asymptotes? Well you might realize that the numerator also equals zero when X is

equal to negative three. What we can do is actually

simplify this a little bit and then it becomes a little bit clear where our vertical asymptotes are. We could say that F of X, we could essentially divide the numerator and denominator by X plus three and we just have to key, if we want the function to be identical, we have to keep the [caveat]

that the function itself is not defined when X is

equal to negative three. That definitely did

make us divide by zero. We have to remember that but that will simplify the expression. This exact same function is going to be if we divide the numerator and denominator by X plus three, it’s going to be three times X minus nine over six times X minus three for X does not equal negative three. Notice, this is an identical definition to our original function and I have to put this

qualifier right over here for X does not equal negative three because our original function is undefined at X equals negative three. X equals negative three is

not a part of the domain of our original function. If we take X plus three

out of the numerator and the denominator, we have to remember that. If we just put this right over here, this wouldn’t be the same function because this without

the qualifier is defined for X equals negative three but we want to have the

exact same function. You’d actually have a

point in discontinuity right over here and now we could think about

the vertical asymptotes. Now the vertical asymptotes

going to be a point that makes the denominator equals zero but not the numerator equals zero. X equals negative three

made both equal zero. Our vertical asymptote,

I’ll do this in green just to switch or blue. Our vertical asymptote is going to be at X is equal to positive three. That’s what made the

denominator equal zero but not the numerator

so let me write that. The vertical asymptote

is X is equal to three. Using these two points of information or I guess what we just figured out. You can start to attempt

to sketch the graph, this by itself is not going to be enough. You might want to also plot a few points to see what happens I

guess around the asymptotes as we approach the two

different asymptotes but if we were to look at a graph. Actually let’s just do it for fun here just to complete the

picture for ourselves. The function is going to

look something like this and I’m not doing it at scales. That’s one and this is

1/2 right over here. Y equals 1/2 is the horizontal asymptote. Y is equal to 1/2 and we have a vertical asymptote that X is equal to positive three. We have one, two … I’m going to do that in blue. One, two, three, once again

I didn’t draw it to scale or the X and Y’s aren’t on the same scale but we have a vertical

asymptote just like that. Just looking at this we don’t know exactly what the function looks like. It could like something like this and maybe does something like that or it could do something like that or it could do something

like that and that or something like that and that. Hopefully you get the idea here and to figure out what it does, you would actually want

to try out some points. The other thing we want

to be clear is that the function is also not defined at X is equal to negative three. Let me make X equals negative three here. One, two, three, so

the function might look and once again I haven’t

tried out the points. It could look something like this, it could look something

where we’re not defined at negative three and then it goes something like this and maybe does something like that or maybe it does something like that. It’s not defined at negative three and this would be an asymptote right now so we get closer and closer and it could go something like that or it goes something like that. Once again, to decide

which of these it is, you would actually want

to try out a few values. I encourage you to, after this video, try that out on yourself and try to figure out

what the actual graph of this looks like.

Where is Mr. Khans Noble Prize God Damn it!!!!!

C'mon, Khan you know that we're all expecting you to make more Bitcoin videos of the other level. How about making some explanatory video in a kids style like (by showing visually limited amount of the Bitcoin money) "Here's the digital blocks that we need to protect with the keys, imagine that this geometrical room is all of the world Bitcoins – now look – if somebody claims to have 1 million of BTC, that takes too much space" – or you can use examples as an earth, to explain that purchasing 100 BTC is like purchasing New York city for cheap 300 years ago.

You know, something like that. We're like a kids – we don't need complicated stuff. At least sometimes make something simple.

Yeah, and I don't know anybody else who can explain Bitcoin geometrically better than you – like linking all the blockchain records together to show movement of money. That kind of video is needed for all those people who's having difficulties of understanding Bitcoin (that's the reason why they don't use it)

If only I had this when I had my last last week!

Why would the function be undefined in two places?

At x=-3 the function value would be 0/0 which is 1 what's wrong with that? It fits into your sketch as far as I can estimate things… Please explain!

You can also find the inverse of f(x) to determine the asymptote.

You must mean x can not be +3 since that is the only vertical asymptote, x = -3 works just fine

Hey Khan, can you finish/add to your differential equations playlist?

You don't have anything on Power Series, Ordinary, Singular and Regular Singular Points or Fourier Series. It'd be great if you could do a bunch on this. Deciphering DiPrima and Boyce's book is a real pain.

I love the tutorial of khan Acady

A parenthesis is missing in the denominator ;)…. great video, as usual!

You are one of the best in the World. Thank you for your idea. It helps to understand the world we living in, and hop that we one day can live without War IN Harmony and Peace ONE DAY. Knowledge is the key for that.

congrats on 4,200th video 😉

You've saved many lives including mine!

Thank you!!

grats on your 4,200 video !

Very helpful, thank you!!

school at its finest! thanks Mr Khan

this problem was a bit more in depth compared to the other tutorials found on youtube but I understood it because of you so thank you!!

Vertical asymptotes at 4:24

Thank you

but wouldn't you have either a hole or a Horizontal Asymtote in an equation ???????????????? please answer?????

Is there sound to this

i love this thing damn

Thank You. Great explanation.

Great vids

Great video, but I couldn't find it anywhere on the actual site no matter where I looked.

Why consider what happens when x approaches negative infinity?

This was extreme helpful but watching video makes me want to swallow my spit when i hear him talk lol

not really helpful when you describe how the first terms will "Dominate". What is that supposed to mean? Please refrain from using buzzwords. I don't have 100% what you are talking about when you talk in this more secluded language.

Oh I hate this stuff! I can't do it. I hate math!

OMG KHAN ACADEMY THANK YOU SOOOOO MUCH FOR MAKING EDUCATIONAL VIDEOS. I GOT 11/10 ON MY ALGEBRA 2 TEST, CONCEPTUAL QUESTIONS PART DUE TO YOUR VIDEOS. I WATCHED YOUR VIDEO AND I AM REALLY HAPPY. THANK YOU SOOO MUCH.

lost me when he got to vertical asymptotes….

Clearly you understand the content but you take forever to explain one question. A good tutor is able to get their point across to their students. An excellent tutor can get their point across both effectively and quickly.

it actually looks like -3 is a vertical asymptote once you graph the original function

The same voice in most of Khan's video…you should be a genius knowing all subjects!

finally!! thanks for the tutorial.

Thank you!!! I have a calculus packet due and a test on Friday, this is a lifesaver!

Thank you so much!! My teacher failed to explain this concept to our class, so I've sent everyone a link to this video. Really, thank you SO much!

for the vertical asymptote, why not just make the denominator's equation equal to zero like: 6x^2-54=0 then do math. it's much more easier.

thanks

vertical asymptote =-3 not 3 and thank U

So what is the significance of vertical asymptotes in practical terms?

yo this helped me out a lot, teacher confuses me prob cause I zone out easily

He teaches an easy topic in the most complicated way possible

1:34 cringe

if someboy doesnt have any concept of this subject,the explanation seems so insufficient.

I couldnt get the points and failed to follow

I dont understand how they can draw on the computer

Wow Really Nice Tutorial, It Really Helped with my Business Cal problem Thank You Khan Academy!

wait do you simplify the equation before using the degree rule to determine the horizontal asymptote?

THANK YOU!!!

So clear – thank you!!!

Why -3 is not in the domain of given equation? You should have explained this.

horizontal asymptotes: what happens if the highest degree terms are not the same on the numerator and denominator?

These two terms are going to dominate….WTF does that mean?

How do u find the y value of the whole discountnity, if i plug neg 3 into x and solve for y it gives my undef

Actually your explanation is complex a bit but it shows all the steps and why this and that become the answer, I suggest you making a video of a lesson with different explanations for different skill learning level and i think this will be very helpful. Thanks a lot for your efforts 👍👏👏👏

so we would say that there is a hole in the graph at x= -3

right?

Please make a video about oblique asymptotes

Vid was helpful

Can he ever pronounce numerator and denominator right?

I love khanacademy and everything, but i feel this could have been explained easier and faster

Sal Khan is a life saver. Much better teacher than Ms. Cornett, who decided it was better to play 2 truths and a lie rather than go over the summer review packet on the first day of school 🙂 And We've got an exam tomorrow :DDDDD. THANKS MS CORNETT

falling out of focus with sal is never ok. thats like day and night.

Khan Academy is the best one as far as I know. Thanks.

I have no clue about the vocabulary but I get the concept so I guess I'm okay.

khan academy is smart….khan academy has brain…thank you mind people!

Thank you Sal for singlehandedly saving my math grade for the last 3 years

Any videos on how to graph something like this?

This is not algebra 2 bro

Is this Algebra 2 or Precalculus?