TABLE OF CONTENTS

4.2 More About Asymptotes

Goals:

• Given a formula for a rational function, the student will be able to name the vertical asymptote(s) of its graph

·         Graph a transformation of a simple rational function using vertical or horizontal shifts

·         Understands the effects of shifting on the asymptotes of the graph

Terms to know:

·         asymptote

·         function

·         domain of a function

·         rational function

Prep assignment

The Effect of Adding a Constant, k

You may recall from section 2.3 that adding a constant to an exponential function resulted in a vertical shift of the original graph.   For example, the graph of  was a vertical shift up 3 of the graph of  The graph of  was a shift down 2 of the graph of .  Let’s explore the effect of adding a constant to a simple rational function.  We begin by comparing a table of values for the reciprocal functions,  and .  First consider , that is, x is approaching positive infinity.

We notice that as ,  seems to be approaching a value of 3, since the fractional part of  is getting smaller and smaller.  Since we are adding 3 to  for each output value, and since  as , we have .  In a similar fashion, as , that is, as x approaches negative infinity,  approaches 3.   (You might complete a table of values for negative values of x if you are not convinced.)   Since this end behavior determines the horizontal asymptote, the horizontal asymptote of the graph of  is .

Now let’s examine what happens to the values  as , that is, as x  approaches 0 from the right (positive) side.  

Since  as , clearly adding 3 to each output will result in .  The chart above confirms this.  Similarly, as , that is, as x approaches 0 from the left (negative) side, we note that .  Thus the vertical asymptote of the graph of  is , the same as it is for the graph of .

In conclusion, the horizontal asymptote of the graph of  is  and the vertical asymptote is .

Checkpoint 4.2 A

Asymptotes for  

The horizontal asymptote of a rational function of the form  is  since as ,  and as , . 

The vertical asymptote is  since as  or ,  or .

Example 1.  Name the asymptotes of the graph of . 

Note that 8 is added on to the output

of ,         

which has  as a horizontal asymptote.

The horizontal asymptote is .

Note that as  or as  ,  or

  .  At , g is undefined.

The vertical asymptote is

.

Example 2.  Name the asymptotes of the graph of  

Note that 4 is subtracted from the output  

of ,  which has  as a horizontal     

asymptote.

The horizontal asymptote is .

Note that as  or as  ,  or

  .  At , h is undefined.

The vertical asymptote is

.

Example 3.  Name the asymptotes of the graph of  

Note that 1 is added to the output

of ,  which has  as a horizontal

asymptote.

The horizontal asymptote is

.

Note that f  as  or as  ,  

or   .  At , r is undefined.

The vertical asymptote is

.

Example 4.  Name the asymptotes of the graph of  

Note that 7 is subtracted from the output of

  ,  which has  as a horizontal

asymptote.

The horizontal asymptote is .

Note that as  or as  ,  or

  .  At , s  is undefined.

The vertical asymptote is

.

Example 5.  Name the asymptotes of the graph of  

Note that 10 is added to the output of        

  ,  which has  as a horizontal

asymptote.

The horizontal asymptote is .

Note that as  or as  ,  or

  .  At , s  is undefined.

The vertical asymptote is

.

Checkpoint 4.2 B

Finding the Domain of a Rational Function

To find the domain of a rational function, consider the values of the input, x, which make the function undefined.  The domain is all real numbers EXCEPT these values.

Example 6.  Find the domain of the function f given by  

Set the denominator equal to 0 and solve:

The domain is all real numbers EXCEPT this

value.

So, the domain is all real numbers except 3.

Write the answer in set notation or as an

 interval

  OR   

Example 7.  Find the domain of the function g given by  

Set the denominator equal to 0 and solve:

The domain is all real numbers EXCEPT 

this value.

So, the domain is all real numbers except -1.

Write the answer in set notation or as an

interval

  OR  

Example 8.  Find the domain of the function h  given by  

Set the denominator equal to 0 and solve:

The domain is all real numbers EXCEPT 

this value.

So, the domain is all real numbers except .

Write the answer in set notation or as an

interval

  OR

Vertical Asymptotes Revisited

Recall from section 4.1 that the vertical asymptote of a reciprocal function, f, given by  or  was the line , the y-axis.  In this section we explore vertical asymptotes of rational functions of the form   and   We first consider the function f given by . 

We noted above that this function is undefined at .  In the chart below, we consider the values of the function as x approaches 3 from the right.  We start with 4 as the input, then gradually move closer and closer to 3.   You might convince yourself that the output values are indeed correct. 

We see that the outputs are increasing without bound, that is, .  If we make a similar chart with the inputs approaching 3 from the left, with x values of  for example, we notice that the outputs are decreasing without bound, that is, .  You might make such a chart and convince yourself that this is true.

So now we have  as  and   as .  Thus, the line  is the vertical asymptote of the graph of f.  We generalize the definition of vertical asymptote from section 4.1:

Vertical Asymptote

Given a constant, h, the line  is a vertical asymptote for a function, f, if as x approaches h, f(x) increases or decreases without bound:

as ,         or        as ,  

If we make similar tables for functions f  of the form , we note that  as  and  as .  Click here for more such examples.

Vertical Asymptote of a Rational Function

For a rational function of the form , the line  is a vertical asymptote.

Thus, the vertical asymptote of the graph of g in example 7 above is .  The function h in example 8 above has 2 vertical asymptotes:   

Checkpoint 4.2 C

Putting it all together:  Graphing a Rational Function

Example 9.  Name the domain and asymptotes of the function h given by .  Find the intercepts, then graph the function and name the range.  Also identify the values of x for which the function is increasing and the values of x for which it is decreasing.

Set the denominator equal to 0 to find  

the domain:

So the domain is  or  

The domain provides us with information

about the vertical asymptote:

 is the vertical asymptote

The end behavior provides information

about the horizontal asymptote:

As  AND as , the line  is the horizontal asymptote.

Find the y-intercept by setting  

, so the point  is on the graph

Find the x-intercept by setting

Finding a few extra points will make our

graph more accurate.   Picking a few

points on either side of the vertical

asymptote(s) is a good idea.

Plotting the above, we obtain:

The graph of  

Note that the range (the set of all outputs) is the set of all real numbers except 0. The function is decreasing over its entire domain, that is it is decreasing over .

Example 10.  Name the domain and asymptotes of the function h given by .   Find the intercepts, then graph the function and name the range.  Also identify the values of x for which the function is increasing and the values of x  for which it is decreasing.

Set the denominator equal to 0 to find the domain:

So the domain is  or  

The domain provides us with information about the vertical asymptote:

 is the vertical asymptote

The end behavior provides information about the horizontal asymptote:

As  AND as , the line  is the horizontal asymptote.

Find the y-intercept by setting  

, so the point  is on the graph.

Find the x-intercept by setting  

So the point  is on the graph. .

Finding a few extra points will make our graph more accurate.  Picking a few points on either side of the vertical asymptote is a good idea.

Plotting the above, we obtain:

The graph of  

Note that the range (the set of all outputs) is the set of all real numbers except -2, that is . The function is decreasing over its entire domain, that is it is decreasing over .

Example 11.  Name the domain and asymptotes of the function h given by .   Find the intercepts, then graph the function and name the range.  Also identify the values of x for which the function is increasing and the values of x for which it is decreasing.

Set the denominator equal to 0 to find the domain:

So the domain is  or  

The domain provides us with information about the vertical asymptote:

 is the vertical asymptote

The end behavior provides information about the horizontal asymptote:

As  AND as , the line  is the horizontal asymptote.

Find the y-intercept by setting  

, so the point  is on the graph.

Find the x-intercept(s) by setting                

Finding a few extra points will make our graph more accurate.  Picking a few points on either side of the vertical asymptote is a good idea.

The graph of  

The range is .  The function is increasing over  and decreasing over .

More worked examples

Homework problems

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