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Asymptotes and Holes in the Graph of Rational Functions

Author: Sophia
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what's covered
In this lesson, you will explore characteristics of rational functions, such as asymptotes and holes in their graphs. Specifically, this lesson will cover:

Table of Contents

1. Finding Vertical Asymptotes and Holes in the Graph of a Rational Function

Consider the function f open parentheses x close parentheses equals 1 over x. Note that the function is undefined when x equals 0 comma meaning the domain of f open parentheses x close parentheses is open parentheses short dash infinity comma space 0 close parentheses union open parentheses 0 comma space infinity close parentheses. Let’s examine the behavior of f open parentheses x close parentheses near x equals 0.

Here is a table of values, which reflects the values of f open parentheses x close parentheses as x gets closer to 0 on the positive side:

bold italic x 0.1 0.01 0.001 0.0001 0.00001
bold italic f open parentheses bold x close parentheses bold equals bold 1 over bold x 10 100 1,000 10,000 100,000

Here is a table of values, which reflects the values of f open parentheses x close parentheses as x gets closer to 0 from the negative side:

bold italic x -0.1 -0.01 -0.001 -0.0001 -0.00001
bold italic f open parentheses bold x close parentheses bold equals bold 1 over bold x -10 -100 -1,000 -10,000 -100,000

Consider the graph of f open parentheses x close parentheses equals 1 over x. 





Now focus on the graph around x equals 0.



Notice how the graph falls indefinitely as x gets closer to 0 from the left side and rises indefinitely as x gets closer to 0 from the right side.



Using arrow notation, we say that f open parentheses x close parentheses rightwards arrow short dash infinity as x rightwards arrow 0 to the power of minus, and f open parentheses x close parentheses rightwards arrow infinity as x rightwards arrow 0 to the power of plus.



If we were to draw the line x equals 0 comma the graph would approach this line as x gets closer to 0 from either side. This line is called a vertical asymptote. 
















big idea





The graph of f open parentheses x close parentheses never crosses a vertical asymptote.







Now consider the function f open parentheses x close parentheses equals fraction numerator 3 x squared minus 12 x over denominator x minus 4 end fraction.



Note that this function is undefined when x equals 4. This means that the domain is open parentheses short dash infinity comma space 4 close parentheses union open parentheses 4 comma space infinity close parentheses. Let’s now examine the behavior of this graph around x equals 4.







bold italic x


3.9


3.99


3.999


4.001


4.01


4.1






bold italic f open parentheses bold x close parentheses bold equals fraction numerator bold 3 bold x to the power of bold 2 bold minus bold 12 bold x over denominator bold x bold minus bold 4 end fraction


11.7


11.97


11.997


12.003


12.03


12.3








Note also that f open parentheses x close parentheses can be simplified by removing a common factor: 	





f open parentheses x close parentheses equals fraction numerator 3 x squared minus 12 x over denominator x minus 4 end fraction equals fraction numerator 3 x open parentheses x minus 4 close parentheses over denominator x minus 4 end fraction equals 3 x comma where x not equal to 4.




The graph of f open parentheses x close parentheses is a line except when x equals 4. Its graph is shown below. 











Notice the hole in the graph at open parentheses 4 comma space 12 close parentheses. The graph has a hole in it since there is no definition for f open parentheses x close parentheses when x equals 4 comma and the graph doesn’t rise or fall indefinitely as x gets closer to 4. 



To summarize, the vertical asymptotes and holes in the graph correspond to the values of x for which the function f open parentheses x close parentheses is undefined. So, how can we tell when the graph of a function will have a vertical asymptote and when there will be a hole in the graph? 
















step by step





Consider a rational function f open parentheses x close parentheses. 

    

  1. Find the real values of x for which f open parentheses x close parentheses is undefined. In other words, the denominator is equal to zero. 
    2. 

  2. If f open parentheses x close parentheses is already in simplest form, then the values found in step 1 correspond to vertical asymptotes. 
    3. 

  3. If f open parentheses x close parentheses can be simplified, then remove common factors. Let g open parentheses x close parentheses equals the simplified form. 
      

    1. Any value of x from step 1 for which the denominator of g open parentheses x close parentheses is equal to zero corresponds to a vertical asymptote of f open parentheses x close parentheses. 
      2. 

    2. Any value of x from step 1 for which the denominator of g open parentheses x close parentheses is not equal to 0 corresponds to a hole in the graph of f open parentheses x close parentheses. The y-coordinate of the hole in the graph is computed by substituting the value of x into g open parentheses x close parentheses.

      3. 

    

    4. 













EXAMPLE

Consider the function f open parentheses x close parentheses equals fraction numerator x squared minus x minus 6 over denominator x squared minus 4 end fraction. Find all vertical asymptotes and/or holes in the graph of f open parentheses x close parentheses. 



First, find all real values of x for which f open parentheses x close parentheses is undefined: 








x squared minus 4 equals 0


Set the denominator equal to 0.






x squared equals 4 
 x equals plus-or-minus 2


Solve.







Next, factor the numerator and denominator of f open parentheses x close parentheses to see if it can be simplified: 








fraction numerator x squared minus x minus 6 over denominator x squared minus 4 end fraction equals fraction numerator open parentheses x minus 3 close parentheses begin display style open parentheses x plus 2 close parentheses end style over denominator open parentheses x plus 2 close parentheses begin display style open parentheses x minus 2 close parentheses end style end fraction


Factor the numerator and denominator.






equals fraction numerator x minus 3 over denominator x minus 2 end fraction


Remove the common factor of open parentheses x plus 2 close parentheses.







Thus, f open parentheses x close parentheses equals fraction numerator x minus 3 over denominator x minus 2 end fraction comma where x not equal to short dash 2. Now, let g open parentheses x close parentheses equals fraction numerator x minus 3 over denominator x minus 2 end fraction.



Note that the denominator of g open parentheses x close parentheses is equal to zero when x equals 2. Since the numerator is not equal to zero when x equals 2 comma this means that x equals 2 is a vertical asymptote of the graph of f open parentheses x close parentheses. 



Since the denominator of g open parentheses x close parentheses is not equal to zero, when x equals short dash 2, there is a hole in the graph of f open parentheses x close parentheses when x equals short dash 2.



The point where the hole is located is open parentheses short dash 2 comma space g open parentheses short dash 2 close parentheses close parentheses equals open parentheses short dash 2 comma space 5 over 4 close parentheses.



In summary, the vertical asymptote is x equals 2 and the hole in the graph is at open parentheses short dash 2 comma space 5 over 4 close parentheses.




















try it





Consider the function f open parentheses x close parentheses equals fraction numerator 3 x over denominator 2 x squared minus x minus 1 end fraction.




Find all vertical asymptotes and/or holes in the graph of f   (x  ).
The factored form is f open parentheses x close parentheses equals fraction numerator 3 x over denominator open parentheses 2 x plus 1 close parentheses open parentheses x minus 1 close parentheses end fraction. Since the numerator and denominator have no common factors, there are no holes in the graph.  



This means that the zeros of the denominator correspond to vertical asymptotes.  






2 x plus 1 equals 0 rightwards arrow x equals short dash 1 half


x minus 1 equals 0 rightwards arrow x equals 1





The equations of the vertical asymptotes are x equals short dash 1 half and x equals 1.


























try it





Consider the function f open parentheses x close parentheses equals fraction numerator 2 x minus 6 over denominator x squared minus 4 x plus 3 end fraction.




Find all vertical asymptotes and/or holes in the graph of f   (x  ).
In factored form, f open parentheses x close parentheses equals fraction numerator 2 open parentheses x minus 3 close parentheses over denominator open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses end fraction.



This means that f open parentheses x close parentheses is undefined when x equals 1 and when x equals 3.  



Now, notice that f open parentheses x close parentheses can be written in simplest form by cancelling out the common factor of open parentheses x minus 3 close parentheses.



If x not equal to 3 comma then f open parentheses x close parentheses equals fraction numerator 2 over denominator x minus 1 end fraction. Now, let g open parentheses x close parentheses equals fraction numerator 2 over denominator x minus 1 end fraction.



Notice that x equals 1 makes both f open parentheses x close parentheses and g open parentheses x close parentheses undefined. This means that there is a vertical asymptote when x equals 1.  



Notice also that x equals 3 makes f open parentheses x close parentheses undefined but g open parentheses x close parentheses is not undefined. This means there is a hole in the graph when x equals 3.



The location of the hole in the graph is open parentheses 3 comma space g open parentheses 3 close parentheses close parentheses.



Since, g open parentheses 3 close parentheses equals fraction numerator 2 over denominator 3 minus 1 end fraction equals 1 comma the graph of f open parentheses x close parentheses has a hole located at open parentheses 3 comma space 1 close parentheses.













In closing, it is possible for a rational function to have neither a vertical asymptote nor a hole in the graph. This occurs when there is no real number x for which the denominator is equal to zero. 







EXAMPLE

Consider the function f open parentheses x close parentheses equals fraction numerator 5 over denominator x squared plus 9 end fraction.



First, note that f open parentheses x close parentheses is in simplest form. 



Next, find all real values of x for which the denominator is equal to zero. 








x squared plus 9 equals 0


Set the denominator equal to 0.






x squared equals short dash 9 
 x equals plus-or-minus 3 i


Solve.







Since there is no real number for which the denominator is equal to zero, there is no vertical asymptote nor a hole in the graph of f open parentheses x close parentheses.




















term to know






Vertical Asymptote


A vertical line x equals a for which f open parentheses x close parentheses rightwards arrow infinity or f open parentheses x close parentheses rightwards arrow short dash infinity as x rightwards arrow a.















2. Finding Horizontal Asymptotes of a Rational Function






Consider the graph of f open parentheses x close parentheses equals 1 over x.











Now focus on the graph as x rightwards arrow short dash infinity and as x rightwards arrow infinity. Notice how the graph appears to “level off” toward y equals 0.



This means that the end behavior of f open parentheses x close parentheses can be described as f open parentheses x close parentheses rightwards arrow 0 as x rightwards arrow short dash infinity and as x rightwards arrow infinity. 



Since y equals 0 is a horizontal line, we say that the graph of f open parentheses x close parentheses has a horizontal asymptote at y equals 0.



Now consider the function f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator x plus 2 end fraction.



Recall that the end behavior of a polynomial is determined by its leading term. In other words: 




    

  • The end behavior of 2 x minus 1 is the same as the end behavior of 2 x. 
    • 

  • The end behavior of x plus 2 is the same as the end behavior of x. 
    • 


This means that the end behavior of f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator x plus 2 end fraction is the same as the end behavior of fraction numerator 2 x over denominator x end fraction equals 2.



This means that f open parentheses x close parentheses rightwards arrow 2 as x rightwards arrow short dash infinity and as x rightwards arrow infinity. 



Thus, y equals 2 is the horizontal asymptote of f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator x plus 2 end fraction.



Before we summarize our findings, another example will be helpful. 







EXAMPLE

Determine the horizontal asymptote of f open parentheses x close parentheses equals fraction numerator 4 x minus 1 over denominator x squared minus x minus 6 end fraction.



Since the numerator and denominator are both polynomials, we can examine each:




    

  • The end behavior of 4 x minus 1 is the same as the end behavior of 4 x. 
    • 

  • The end behavior of x squared minus x minus 6 is the same as the end behavior of x squared. 
    • 


This means that the end behavior of f open parentheses x close parentheses equals fraction numerator 4 x minus 1 over denominator x squared minus x minus 6 end fraction is the same as the end behavior of fraction numerator 4 x over denominator x squared end fraction comma or 4 over x.



But, as x rightwards arrow short dash infinity or x rightwards arrow infinity comma 4 over x rightwards arrow 0. (Dividing 4 by a very large number results in a number close to 0). 



Thus, y equals 0 is the horizontal asymptote of f open parentheses x close parentheses equals fraction numerator 4 x minus 1 over denominator x squared minus x minus 6 end fraction.


 




This means that there are certain conditions under which f open parentheses x close parentheses has a horizontal asymptote. Here is a summary. 
















big idea





Consider a rational function f open parentheses x close parentheses equals fraction numerator N open parentheses x close parentheses over denominator D open parentheses x close parentheses end fraction comma where N open parentheses x close parentheses and D open parentheses x close parentheses are polynomials. 



If the degree of N open parentheses x close parentheses is smaller than the degree of D open parentheses x close parentheses comma then the horizontal asymptote is y equals 0.



If the degrees of N open parentheses x close parentheses and D open parentheses x close parentheses are equal, then the horizontal asymptote is y equals fraction numerator leading space coefficient space of space N open parentheses x close parentheses over denominator leading space coefficient space of space D open parentheses x close parentheses end fraction.



If the degree of N open parentheses x close parentheses is larger than the degree of D open parentheses x close parentheses comma then f open parentheses x close parentheses has no horizontal asymptote.







Here is an example to illustrate this important idea.







EXAMPLE

Consider the functions f open parentheses x close parentheses equals fraction numerator 12 x over denominator x cubed plus x squared plus 4 end fraction and g open parentheses x close parentheses equals fraction numerator 3 x squared plus 8 over denominator 5 x squared plus x plus 3 end fraction.



The function f open parentheses x close parentheses has horizontal asymptote y equals 0 since the numerator has degree 1, which is less than the degree of the denominator, which is 3.



The function g open parentheses x close parentheses has horizontal asymptote y equals 3 over 5. Since the degrees of the numerator and denominator of g open parentheses x close parentheses are the same (they are both 2), the leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 5.




















try it





Consider the functions f open parentheses x close parentheses equals fraction numerator x squared plus 4 over denominator x cubed plus 8 end fraction and g open parentheses x close parentheses equals fraction numerator 1 minus 3 x over denominator 7 x plus 2 end fraction.




Write the equation of the horizontal asymptote of f   (x  ).
Since the degree of the numerator is less than the degree of the denominator, the equation of the horizontal asymptote is y equals 0.











Write the equation of the horizontal asymptote of g   (x  ).
Since the degrees in the numerator and denominator are the same, the equation of the horizontal asymptote is determined by the ratio of the leading coefficients:  




y equals short dash 3 over 7


























term to know






Horizontal Asymptote


A horizontal line y equals b for which f open parentheses x close parentheses rightwards arrow b as x rightwards arrow short dash infinity or x rightwards arrow infinity.















3. Finding Slant and Nonlinear Asymptotes of a Rational Function






Consider a rational function f open parentheses x close parentheses equals fraction numerator N open parentheses x close parentheses over denominator D open parentheses x close parentheses end fraction where N open parentheses x close parentheses and D open parentheses x close parentheses are polynomials.



When the degree of N open parentheses x close parentheses is greater than the degree of D open parentheses x close parentheses by one, then the end behavior of f open parentheses x close parentheses is described by a slant asymptote. 



When the degree of N open parentheses x close parentheses is greater than the degree of D open parentheses x close parentheses by more than one, then the end behavior of f open parentheses x close parentheses is described by a nonlinear asymptote. 



Both types are found by performing long division or when appropriate, synthetic division. 



Before finding these asymptotes, let’s review the division algorithm for polynomials. 



By the division algorithm, we know that fraction numerator N open parentheses x close parentheses over denominator D open parentheses x close parentheses end fraction has quotient Q open parentheses x close parentheses and remainder R open parentheses x close parentheses for which N open parentheses x close parentheses equals D open parentheses x close parentheses Q open parentheses x close parentheses plus R open parentheses x close parentheses. Now, let’s rewrite the equation.







N open parentheses x close parentheses equals D open parentheses x close parentheses Q open parentheses x close parentheses plus R open parentheses x close parentheses


This is the original equation.






fraction numerator N open parentheses x close parentheses over denominator D open parentheses x close parentheses end fraction equals fraction numerator Q open parentheses x close parentheses D open parentheses x close parentheses over denominator D open parentheses x close parentheses end fraction plus fraction numerator R open parentheses x close parentheses over denominator begin display style D end style open parentheses x close parentheses end fraction


Divide both sides by D open parentheses x close parentheses.






f open parentheses x close parentheses equals Q open parentheses x close parentheses plus fraction numerator R open parentheses x close parentheses over denominator D open parentheses x close parentheses end fraction


Simplify and replace fraction numerator N open parentheses x close parentheses over denominator D open parentheses x close parentheses end fraction with f open parentheses x close parentheses.








Remember that according to the division algorithm, the degree of R open parentheses x close parentheses is less than the degree of D open parentheses x close parentheses. This means that fraction numerator R open parentheses x close parentheses over denominator D open parentheses x close parentheses end fraction rightwards arrow 0 as x rightwards arrow short dash infinity and as x rightwards arrow infinity.



Thus, the end behavior of f open parentheses x close parentheses is the same as the end behavior of Q open parentheses x close parentheses comma the quotient.



A slant asymptote is the oblique asymptote obtained when the degree of the numerator is one greater than the degree of the denominator. 







EXAMPLE

Consider the function f open parentheses x close parentheses equals fraction numerator x squared minus 2 x minus 8 over denominator x minus 3 end fraction.



Since the degree of the numerator is one more than the degree of the denominator, f open parentheses x close parentheses has a slant asymptote. To find this, we use division. 



Notice that the denominator is x minus 3. Since this is a linear term with leading coefficient equal to 1, synthetic division can be used. 








table attributes columnalign left end attributes row cell table row 3 row blank end table stack open vertical bar table attributes columnalign right end attributes row 1 cell short dash 2 end cell cell short dash 8 end cell row blank blank blank end table close with bar below end cell row cell space space space space space space 1 end cell end table


Set up the synthetic division.






table attributes columnalign left end attributes row cell table row 3 row blank end table stack open vertical bar table attributes columnalign right end attributes row 1 cell short dash 2 end cell cell short dash 8 end cell row blank 3 3 end table close with bar below end cell row cell space space space space space space 1 space space space space space 1 space space short dash 5 end cell end table


Perform the synthetic division.







With remainder -5, the quotient is x plus 1. This means that the equation of the slant asymptote is y equals x plus 1.




















try it





Consider the function f open parentheses x close parentheses equals fraction numerator x squared minus 2 over denominator x minus 4 end fraction. 




Write the equation of its slant asymptote.
Since the denominator is a linear factor, we can find the quotient using synthetic division. Here is the table: 



table row 4 row blank end table stack open vertical bar table attributes columnalign right end attributes row 1 0 cell short dash 2 end cell row blank cell space 4 end cell 16 end table close with bar below
space space space space space space 1 space space space 4 space space space 14



This means that f open parentheses x close parentheses equals x plus 4 plus fraction numerator 14 over denominator x minus 4 end fraction.



The slant asymptote is f open parentheses x close parentheses without the remainder term, which means the equation of the slant asymptote is y equals x plus 4.













When the denominator doesn’t have the form x minus k comma long division is used. 







EXAMPLE

Consider the function f open parentheses x close parentheses equals fraction numerator 4 x cubed minus 11 x minus 3 over denominator 2 x minus 1 end fraction.



The slant asymptote is found by using long division. Here is the long division worked out: 




table attributes columnalign left end attributes row cell space space space space space space space space space space space space space space space space space space space space space space 2 x squared space space plus space space x space minus 5 end cell row cell 2 x minus 1 long division enclose 4 x cubed plus 0 x squared minus 11 x minus 3 end enclose end cell row cell space space space space space stack space space minus open parentheses 4 x cubed minus 2 x squared close parentheses with bar below end cell row cell space space space space space space space space space space space space space space space space space space space space space space 2 x squared minus 11 x end cell row cell space space space space space space space space space space space space space space space stack space space minus open parentheses 2 x squared space space minus space space x close parentheses with bar below end cell row cell space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space short dash 10 x minus 3 end cell row cell space space space space space space space space space space space space space space space space space space space space space space space stack space space minus open parentheses short dash 10 x plus 5 close parentheses with bar below end cell row cell space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space short dash 8 end cell end table



This means the equation of the slant asymptote is y equals 2 x squared plus x minus 5.




















watch





Finding asymptotes and holes in the graphs can often be a tricky process. This video gives you another opportunity to investigate these characteristics of the function f open parentheses x close parentheses equals fraction numerator x cubed minus 2 x squared minus 11 x plus 12 over denominator x squared minus 1 end fraction.














 















try it





Consider the function f open parentheses x close parentheses equals fraction numerator x cubed minus 8 over denominator x squared plus 2 x plus 5 end fraction.




Write the equation of the slant asymptote of f   (x  ).
Since the denominator is quadratic (not linear), we use long division to determine the equation of the slant asymptote. 



space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space x minus 2
x squared plus 2 x plus 5 long division enclose x cubed plus 0 x squared plus 0 x minus 8 end enclose
space space space space space space space space space space space space space space space space space space stack negative open parentheses x cubed plus 2 x squared plus 5 x close parentheses with bar below
space space space space space space space space space space space space space space space space space space space space space space space space space space space space short dash 2 x squared minus 5 x minus 8
space space space space space space space space space space space space space space space space space space space space space space space stack negative open parentheses short dash 2 x squared minus 4 x minus 10 close parentheses with bar below
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space short dash x plus 2



Therefore, f open parentheses x close parentheses equals x minus 2 plus fraction numerator short dash x plus 2 over denominator x squared plus 2 x plus 5 end fraction comma which means the equation of the slant asymptopte is y equals x minus 2.


























terms to know






Slant Asymptote


The asymptote obtained when the degree of the numerator is one greater than the degree of the denominator. This means that the graph of a rational function approaches a linear function as x rightwards arrow short dash infinity and x rightwards arrow infinity.



Nonlinear Asymptote


The asymptote obtained when the degree of the numerator is more than one greater than the degree of the denominator. This means that the graph of a rational function approaches a nonlinear function (in our case, a polynomial function) as x rightwards arrow short dash infinity and x rightwards arrow infinity.



















summary






In this lesson, you learned that graphs of rational functions have several important characteristics. Values of x that make f open parentheses x close parentheses undefined could result in vertical asymptotes or holes in the graph of a rational function. You also learned that the horizontal, slant, and nonlinear asymptotes describe the end behavior of a rational function. The graph of a rational function has either a horizontal asymptote, a slant asymptote, or a nonlinear asymptote; it is not possible for it to have more than one of these types.








SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.



















Terms to Know









Horizontal Asymptote





A horizontal line y equals b for which f open parentheses x close parentheses rightwards arrow b as x rightwards arrow short dash infinity or x rightwards arrow infinity.





Nonlinear Asymptote





The asymptote obtained when the degree of the numerator is more than one greater than the degree of the denominator. This means that the graph of a rational function approaches a nonlinear function (in our case, a polynomial function) as x rightwards arrow short dash infinity and x rightwards arrow infinity.





Slant Asymptote





The asymptote obtained when the degree of the numerator is one greater than the degree of the denominator. This means that the graph of a rational function approaches a linear function as x rightwards arrow short dash infinity and x rightwards arrow infinity.





Vertical Asymptote





A vertical line x equals a for which f open parentheses x close parentheses rightwards arrow infinity or f open parentheses x close parentheses rightwards arrow short dash infinity as x rightwards arrow a.
































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