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# Graphing rational functions 4

Sal graphs y=(x)/(x^2-x-6). Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

• Can an asymptote have an asymptote or a point where it doesn't exist?
• Interesting question, but I can't think of any way for either of those to happen.

The asymptotes you will typically see at this level are all lines (horizontal, vertical, or oblique/slant). Curved asymptotes do exist, but they are 'simple' polynomials not rational equations. This means that there is no denominator and so no way for division by zero to occur.

http://www.purplemath.com/modules/asymtote3.htm
(1 vote)
• The rational function that I'm graphing has an expression with three terms in the numerator and the denominator. I probably won't get a response in time, but for future reference; how would I go about with graphing this sort of rational function?
• Well, first see if you can get into factored form. If you can combine the three terms into factors of terms, you should be all set as you know the locations of the asymptotes and everything. If you can't factor anything, you can at least determine the asymptotes of the function by applying the rational roots theorem to the denominator. You can also find the horizontal/slant asymptote by looking at the degrees.
• how can the graph cross 0 if the horizontal asymptote is at 0?
• Graphs of rational functions can cross the horizontal asymptote because it may not be undefined at that value But, the graph will never cross a vertical asymptote. The vertical asymptotes occur where the function is undefined.
• How do I find the horizontal asymptote if the numerator does not contain an x?
• If the numerator is a constant and the denominator is a polynomial then the asymptote will always be at `y=0`.
• Why can the parts of the function at the far negative and far positive portions of the graph not pass through the horizontal asymmtote, while the part in the middle can? Also, how do you know whether a part of the function will pass through the horizontal asymmtote?
Thanks!
• The horizontal asymptote is not much like a vertical one, It's caused by trends as x gets very large, not by /0. So before |x| gets large things can be very different.
Just plot the graph according to the methods described so far and see where the points lie. Whether or not a function passes through a horizontal asymptote depends on the function.
• If a point crosses an asymptote, is that a point of discontinuity?
• I'm not sure what you mean by a point crossing an asymptote – do you mean a line?

The graph of a function can't cross a vertical asymptote, and thus vertical asymptotes are a type of discontinuity.

The graph of a function can cross a horizontal asymptote – no discontinuity.

Except for piecewise functions, you only get discontinuities when there is division by zero.
(1 vote)
• If y=0 is an asymptote, how is there a value for x at y=0? Shouldn't an asymptote mean the graph is not touching that line at all?
• y=0 is an asymptote only when x<-2 and x>3.
(1 vote)
• In all of the videos you explained how to find asymptotes when we have "something / something",but what"s the approach if i have "something +(something/something)",maybe something like this y=2x+(1/x-1),how to find the asymptotes of that? If the problem was 1/x-1 OK,similar to your examples , but how to approach it now?
(1 vote)
• You just get the common denominator;
2x+(1/(x-1)) = (2x^2 - 2x)/(x-1)
Now you set the denominator equals zero;
x-1 = 0 -> x=1

Now, you set the limit to x->infinity;
lim y where x -> ± infinity =
lim x -> ± infinity = 2x + (1/(x-1)) = 2x
- Because, if you imagine that 1/x-1, and x was infinite large, it would be almost 0, right?

There you have it;
Diag. asymptote: Y= 2x
vertical asymptote: x=1
• What happens when the numerator has higher degree term than the denominator ?
(1 vote)
• You have to divide the numerator by the denominator. Either long division or synthetic division, there won't be a horizontal asymptote, the asymptote will be oblique. Note, that there is an oblique asymptote only if the numerator degree is 1 greater than the degree of the denominator, if the degree is greater then 1, then there is no asymptote.