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4.2.   More Worked Examples:  Graphing a Rational Functions

Example 1.  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.

x

-9

-4

-2

3

9

y

-1

-6

6

1

1/2

 

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 2.  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.

x

-3

-1

.5

2

y

4.7555-1.5

4.5

3

6

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 5, that is . The function is decreasing over its entire domain, that is it is decreasing over .

 

 

 

 

Example 3.  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.

x

-7

-6

-4

-3

y

-3.75

-3

-3

-3.75

 

The graph of  

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

 

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