What is an asymptote and why doesn't parabola have one?

[Skhandelwal]

The thing I have noticed is that parabola's angle from the x-axis keeps increasing...meaning the object is decreasing from that direction...if that is the case, then how does it go on in that direction forever even though, its speed going in that direction keeps decreasing?

At what rate something is decreasing if it has a parabola?(Plz don't explain it in formula, I know that explanation too...I just want more conceptual explanation) I know when x is in the denominator, that when there are 2 asymptotes.

 

[Eighty]

A function f(x) has an asymptote if it "looks like" a straight line when x tends to \pm\infty. More precisely, if the distance |f(x)-\text{the line}| tends to 0.

A parabola doesn't have one. Suppose there was one (draw an arbitrary line in the same graph). Then, at some point, the parabola will rush past that line and keep going away from it, so it wasn't an asymptote.

 

[Skhandelwal]

Thx for that approach but I guess I wasn't really looking for a counter-example...more like the real WHY? Which is based w/o math...like a drawing...if someone drew an infinite parabola, how can it approach infinity toward the x-axis...b/c the rate at which it is approaching x-axis is decreasing.

Btw, this brings the question, at what rate is something deaccelerating if a graph has an asymptote? For instance: y=1/x

 

[HallsofIvy]

Thx for that approach but I guess I wasn't really looking for a counter-example...more like the real WHY? Which is based w/o math...like a drawing...if someone drew an infinite parabola, how can it approach infinity toward the x-axis...b/c the rate at which it is approaching x-axis is decreasing.

Btw, this brings the question, at what rate is something deaccelerating if a graph has an asymptote? For instance: y=1/x

The words "accelerating" and "decelerating" have nothing to do with parabolas. If you can phrase the question purely in terms of mathematics rather than physics we might be able to understand it better.

The answer to your first question is really in what you said: "parabola's angle from the x-axis keeps increasing". Since it does not approach anyone angle, it cannot have an asymptote. The graph of y= x² "can approach infinity toward the x-axis" because x² is defined for all x, not matter how large!

 

[Skhandelwal]

But it IS approaching an angle! 180 degrees from the x-axis.

 

[neg_ion13]

I wouldn't look at it in terms of increasing angles. I think that will lead to trying to visually evaluate your question. Think of more like, for each value of X, Y gets much bigger than X. An asymptote is a place where a function is undefined. X^2 is never undefined so it can not have an asymptote. In a nut shell there is not a value of X that can't be squared, so if X can always be squared then any value of X can be used, therefore X can approach infinity.

 

[Skhandelwal]

Well yeah, but as I mention before...I was looking for NONMATHEMATICAL example. It is similar to like this...the doctor tells me, I am dead...but I say...I clearly don't feel dead...he says...well...the machines indicate you are dead so you are dead.

I mean I see that MATH indicate it will go to infinity. But when I look it from my eyes, that is just not what I see. And I just wish there was a more satisfying way to understand this.

 

[neg_ion13]

You can't define something approaching infinity by "looking at it". What you think is common sense may not be quite how it works. Also, I did not give you a mathematical example. There were no calulations in what I posted. It goes to infinity is the answer your looking for. If it doesn't stop how can it approach an asymptote?

 

[codec9]

I see the problem. It is right to say that the angle of the section you are looking at will keep increasing with respect to the x-axis. If I understand this correctly, the angle will approach 90 degrees. 180 degrees would mean it is parallel to the x-axis if you are measuring with respect to it.

You can look at it from the angles all you want and that is perfectly correct. However, while every function that does have an asymptote does approach an angle just like the parabola, that is not sufficient grounds to conclude that IF a function approaches an angle THEN it is asymptotic. It's true that asymptotic functions do that. But it is not true that all functions that do that are asymptotic. Tricky, tricky.

The way I like to break it down is this. There is a group of functions that approach a given angle. Some of them are asymptotic, but not all of them. Why? Because an asymptote means the functions can never quite get to some value or other that it gets real close to, and as a result has this angle business. But for a parabola, the function can take in any x value and output any y value. (Not every x is paired with every y.) Actually, it only outputs every y value equal or greater than the bottom of the parabola, of equal to or less than the top if it is pointing down.

The point is: the function must lack the ability to reach a certain VALUE and approach that value arbitrarily closely. I think the difficulty is that, although the parabola gets steeper and steeper, it still keeps going. You just have to back up and "look at" more.

What might break your spell is this. The parabola can take in ANY x value as its input. The height is the x value squared. Where is the y value? Well, it is straight above the x! So if we can do this for any x value, then we might as well be plotting y = x because that does the same thing. It takes in a "width" and outputs a "height" directly above it. Alas, the x in y = x is not squared so the heights are not as high, but that does not matter. That has to do with the shape not the relationship I am talking about here (how y's get mapped from the x).

If there is a shift left or right, it goes over that many units so its no problem. Just shift your axes and you are back to the original. Same for up or down.

To repeat, anywhere on the x-axis you choose, there is a y some distance above the x. The thing is not limited by width and certainly not by height, although it can't go above/below the vertex or whatever it's called.

 

[Skhandelwal]

OMG, someone finally understood my question! lol, but still doesn't solve the problem...the thing is...I like to look at it as if it is similar to that paradox...here it is

Paradox: Let's say you want to travel 1 meter, what you do is that you travel half the distance: .5 meter, then travel half the distance of what you have already traveled:.5 and keep repeating the process infinitely so you keep getting closer, but never get there.

Based on this paradox...I believe that if it is keep approaching 90 degrees angle...even though it is getting closer to infinity, it can never achieve it conceptually..get it?

 

[codec9]

infinity has to do with height, not angle. y = x goes to infinity in either direction, but it is a clear 45 degrees with respect to the x-axis no matter where you are.

 

[CRGreathouse]

OMG, someone finally understood my question! lol, but still doesn't solve the problem...the thing is...I like to look at it as if it is similar to that paradox...here it is

Paradox: Let's say you want to travel 1 meter, what you do is that you travel half the distance: .5 meter, then travel half the distance of what you have already traveled:.5 and keep repeating the process infinitely so you keep getting closer, but never get there.

Based on this paradox...I believe that if it is keep approaching 90 degrees angle...even though it is getting closer to infinity, it can never achieve it conceptually..get it?

That's Zeno's paradox of the arrow, yes? The resolution is that you *can* travel an infinite number of steps infinite time, provided the total distance is finite (and sufficiently small).

Because the parabola has an increasing slope (= angle from the x-axis) it cannot have an asymptote other than a vertical asymptote. If the parabola had a vertical asymptote then there would be some point where you could get infinitely large values as you became arbitrarily close to that point, but there is no such point on the parabola. Take y=x^2 as an example; for every x the limit as x approaches k of y is just k^2, which is finite; it's not unbounded in any finite interval.

 

[Skhandelwal]

See that's the point...all the explanations that people have been giving me have to do w/ mathematical calculations...why can't someone just teach me the concept WHY it does approach infinity?

 

[MaWM]

Ok. no sweat. Its because x^2 is defined for all x. x^2 gets bigger and bigger, faster and faster, but there is no x where it gets infinitely large. It keeps getting steeper and steeper. Its slope approaches verticality. But that's not enough to have an asymptote. You have to approach vertically *fast enough*. x^2 does not. it takes infinitely many x s before x^2 becomes vertical.

If you forgive the mathematical example, consider the series SIGMA(1/n) . Each step is smaller than the last. The size of the terms goes to zero, and yet their sum is unbounded. What does this have to do with the parabola? Well, if you look at the distance it takes for the slope of the parabola to double, that distance gets smaller and smaller as the x gets larger. But it still takes infinitely large x until the slope goes to infinity.

Other functions, like 1/x, do have asymptotes because their slope increases much more quickly than x^2

 

[stewartcs]

An asymptote is a straight line associated with a curve such that as a point moves along an infinite branch of the curve the distance from the point to the line approaches zero and the slope of the curve at the point approaches the slope of the line.

The word's origin is Greek and means "not intersecting". It's used in mathematics to merely describe an imaginary line that a curve cannot cross.

Is that conceptual enough?

 

[Skhandelwal]

LOL...did you even read my question? I am not asking what an asymptote is...I am asking...why doesn't parabola has a verticle one.

btw,(this is a mathematical question :-) at what rate is something decreasing if it has an asymptote?

 

[stewartcs]

What is an asymptote and why doesn't parabola have one?

Is this not the title of this thread??

 

[stewartcs]

See that's the point...all the explanations that people have been giving me have to do w/ mathematical calculations...

btw,(this is a mathematical question :-) at what rate is something decreasing if it has an asymptote?

So first you don't like the mathematical answers, but then you admit its a mathematical question??

 

[stewartcs]

I am not asking what an asymptote is...I am asking...why doesn't parabola has a verticle one.

A parabola is a continuous function. That means it doesn't have any discontinuities. An asymptote creates a discontinuity in the would be path of a parabola.

btw,(this is a mathematical question :-) at what rate is something decreasing if it has an asymptote?

An asymptote is a straight line; its rate of change is zero.

 

[Hurkyl]

Consider the graph of y = 10 / (x^2 + 1).

Is the line y=0 an asymptote?
Is the line y=1 an asymptote?
Is the line y=-1 an asymptote?

(The answers are not all the same)


Incidentally, projectively, asymptotes are simply tangent lines: they are tangent at infinity. A parabola has only one point at infinity, and the tangent line there is the line at infinity.

 

[Skhandelwal]

Ok...there seems to be a huge misunderstanding...first of all, stewartcs, I apologize for how I reacted to your answer...I got so involved in the parabola...I forgot about that...about the asymptote...notice...I am talking about a PARABOLA...so my question is...at what rate is the PARABOLA decreasing when it has an asymptote...notice here, I am not asking the rate at which the asymptote is changing...but the parabola's rate which is being affected by the asymptote...now...this question was suppose to be mathematical b/c giving a conceptual answer to this seems a bit difficult and unneccesary. So for the rate...I want some number...but for the question:Why doesn't parabola have an asymptote...I need a conceptual answer...Is it clearer now?

I sure do hope so.

 

[stewartcs]

at what rate is the PARABOLA decreasing when it has an asymptote

Parabolas do not have asymptotes, therefore your question is nonsensical. It's like saying why is a circle square?

If you are instead asking what is the rate of change at any local vertex of a parabola, then the answer is zero.

 

[Skhandelwal]

My bad again...lemme rephrase the question...at what rate does a FUNCTION decrease when it has an asymptote?(mathematical answer)

Why doesn't parabola have an asymptote?(conceptual answer-don't give me a counter argument...meaning...don't tell me if it would...it would be this kinda function...stuff like that...but tell me the actual WHY.)

 

[codec9]

stewart, regarding post 19 the rate of change of a line is not automatically 0. I don't know where you got that from. A vertical line has an infinite rate of change; y = x has a rate of change of 1.

Skhandelwal, was there something wrong with my explanations on posts 9 and 11?
MaWM's first paragraph in post 14 is the concept you are looking for I think. So a function is just a rule basically. It might be easier to describe by talking about sets of stuff. You start with a set of numbers and do something to them and end up with another set of numbers. This is not really mathematical. In the case of the parabola, you start with all the numbers from -infinity to +infinity and, for each, you times it by itself and you end up with that corresponding set.

If you agree that what goes into the function goes to infinity in either direction, and all we do to each of those things is square it and therefore make it bigger in absolute value (except for fractions), how do you not see that the bigger thing is not going to infinity as well? The bigger thing gets to infinity faster!

Now for it not to have an asymptote the outputs can't just go to infinity, because that can happen with a vertical asymptote like in 1/x, but the real trick is that the output is right above the input, and the inputs are from -infinity to +infinity, combined with the outputs not being limited as well. So there is not any sort of asymptote or "line where the function arbitrarily closely approaches without getting there."

A function can decrease or increase at any rate and have an asymptote. The asymptote does not have to be vertical, it could be y = 3x, in which case the function would be increasing at the rate of 3.

 

[MaWM]

ok.. first off. there are polynominals. X^2 increases quickly, but not quickly enough. x^100 increases far more quickly, but still not quickly enough (to have an asymptote). Beyond any polnominal, there are functions of the form 2^x. These increase more quickly than *any* polynomial, no matter how high the power. But they *still* don't increase quickly enough to have an asymptote. The simplest example of a function that does have an asymptote is 1/x . this function increases even more quickly than any function of the form n^x or n^n^x or n^n^n^x etc. Every function that increases more quickly than 1/x does have at least one asymptote. 1/x^2, 1/x^100 etc

 

[Hurkyl]

My bad again...lemme rephrase the question...at what rate does a FUNCTION decrease when it has an asymptote?(mathematical answer)

It doesn't have to decrease at all. For example, take the function
f(x) = \frac{x^3 + x + 1}{x^2 + 1}.
(I think I have that right)

Or consider f(x) = -1/x.


P.S. functions don't have asymptotes. I'm assuming you meant "when its graph has an asymptote".

 

[symbolipoint]

What is an asymptote? Read this:
http://en.wikipedia.org/wiki/Asymptote

Other than for lines perpendicular to axis of symmetry and 'on the other side of the vertex', No line exists such that when a value of the independant variable is infinitessimally approached, a dependant variable is not reached on (for graph of a parabola, and assumed rotated into standard horizontal or vertical position).

Knowledgable readers could check what is stated above for reliablility and make adjustments.

 

[stewartcs]

stewart, regarding post 19 the rate of change of a line is not automatically 0

Yeah I suppose I should have been more specific...I was referring to a horizontally straight line. Thanks for pointing that out codec9.

 

[futurebird]

For f(x)=x^2

You can pick an x, as large as you like and the curve will always be above that x. It may be WAY above the x, but it's up there... somplace. No matter what.

That's how I think of it.

 

[Skhandelwal]

MaWM, you are on to something...even though you explaining it mathematically, you are getting the point...my question.

You are saying how it doesn't approach infinity "fast enough"...but why does speed come into play? I mean as long as something is approaching the slope undefined...how can it ever approach infinity? Wouldn't the slope of the parabola become undefined at the point infinity?

Am I being clear?

 

[codec9]

The slope of x^2 is 2x. Therefore, as x goes to infinity so does the slope.

 

[MaWM]

MaWM, you are on to something...even though you explaining it mathematically, you are getting the point...my question.

You are saying how it doesn't approach infinity "fast enough"...but why does speed come into play? I mean as long as something is approaching the slope undefined...how can it ever approach infinity? Wouldn't the slope of the parabola become undefined at the point infinity?

Am I being clear?

Yes, clear enough. We generally don't think about asymptotes at infinity. Asymptotes are generally reserved for functions that reach infinity at a *finite* value of x. I suppose you can think of a parabola as having something like an asymptote at infinity, but that description pretty much violates the requirements of the definition of an asymptote. And.. in order to go to infinity at a finite x, you have to be increasing "fast enough". (As I am sure you can appreciate, reaching infinity over finite x, is much more difficult than acheiving it in infinite x)

 

[HallsofIvy]

No, we often talk about "horizontal asymptotes". y= 1/x has y=0 as horizontal asymptote.
y= \frac{x^3- 1}{x+ 1} has the line y= x as an asymptote at infinity because for very large x, x³- 1 is approximately x³ and x+ 1 is approximately x.

 

[Skhandelwal]

Again, I still don't get "fast enough" I mean I know what you mean by it...but mathematically, what is the rate at which something decreases which makes it have an asymptote? Personally, I feel an acceleration will result in it failing the vertical line test and thus qualifying it as nonfunction. So how do we describe that rate? Is that some derivative of acceleration? If yes then which? This is all very confusing.

 

[codec9]

There is no certain rate. Read the thread, or at least the post above yours!

 

[MaWM]

Again, I still don't get "fast enough" I mean I know what you mean by it...but mathematically, what is the rate at which something decreases which makes it have an asymptote? Personally, I feel an acceleration will result in it failing the vertical line test and thus qualifying it as nonfunction. So how do we describe that rate? Is that some derivative of acceleration? If yes then which? This is all very confusing.

Nope, an increasing slope is not enough. It will still take infinitely many xs before the slope is vertical. Nor is having the slope increasing at an accelerting rate enough. In fact, even if *every* derivative of the function is increasing, that is still not fast enough. (functions like 2^x having increasing derivatives of all orders). Its hard to describe how fast a function like 1/x increases before it has an asymptote. Not only does the function itself become undefined at 0, but all of its derivatives do too. That is to say, not only does the magnitude of 1/x grow infinity fast near zero, but the magnitudes of all of its derivatives also grow infinitly fast. So, the kind growth required to have an asymptote is fast beyond the capability of being describable by any sort of derivative.

 

[Skhandelwal]

hmm...codec 9, For some reason, even though I understand the Math of Halls of Ivy, I am having a hard time understanding his point.(Damn that dumb head of mine!)

MaWM, I love your simplicity. However, you are missing a point. It can not really be described in terms of acceleration b/c if the x values were decreasing at a certain rate, they would start going back, and thus start repeating the x values they have already been in. So, if it is not an acceleration, and is beyond all its derivate like you said...how can this be described?(lets not bring summation formulas into this)

But is there another way?

 

[Hurkyl]

So, if it is not an acceleration, and is beyond all its derivate like you said...how can this be described?

As having an asymptote.