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4.1.   More Worked Examples:  Graphs of Simple Rational Functions

Characterize the graph of each of the following by naming

a)    the domain

b)   the end behavior:  As  and as  

c)    the asymptotes

d)   the intercepts

 

Use this information to sketch the graph by hand , finding at least one additional point on either side of the vertical asymptote.  Then name

e)    the range and

f)       the interval(s) over which the function is increasing and/or decreasing. 

.

 

Example 1.   

Set the denominator = 0 to find the values for which the function is undefined.  The domain is all real numbers EXCEPT this value.

 

 is the domain

Choose some very large values for x and some very small values for x to determine end behavior.

 

As  

As  

The vertical asymptote(s) occurs at the value(s) for which the function is undefined.

 

The vertical asymptote is  

The end behavior determines the horizontal asymptote.

The horizontal asymptote is  

 

To find the y-intercept, set  

, which is undefined.  There is no

               y-intercept.

To find the x-intercept(s) , set  

 

 

 

Thus, there are no x-intercepts

 

Finding a few extra points makes our graph more accurate.

x

1

2

3

-3

-3

y

6

3

2

-2

-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.   

 

Set the denominator = 0 to find the values for which the function is undefined.  The domain is all real numbers EXCEPT this value.

 

 is the domain

Choose some very large values for x and some very small values for x to determine end behavior.

 

As  

As  

The vertical asymptote(s) occurs at the value(s) for which the function is undefined.

 

The vertical asymptote is  

The end behavior determines the horizontal asymptote.

The horizontal asymptote is  

 

To find the y-intercept, set  

, which is undefined.  There is no

               y-intercept.

To find the x-intercept(s) , set  

 

 

 

Thus, there are no x-intercepts.

 

Finding a few extra points makes our graph more accurate.

x

  1

  2

  4

-1

-2

-4

y

-8

-4

-2

 8

 4

 2

 

 

 

 

Plotting the above, we obtain:

 

The graph of  

 

Example 3.   

 

Set the denominator = 0 to find the values for which the function is undefined.  The domain is all real numbers EXCEPT this value.

 

 is the domain

Choose some very large values for x and some very small values for x to determine end behavior.

 

As  

As  

The vertical asymptote(s) occurs at the value(s) for which the function is undefined.

 

The vertical asymptote is  

The end behavior determines the horizontal asymptote.

The horizontal asymptote is  

 

To find the y-intercept, set  

, which is undefined.  There is no

               y-intercept.

To find the x-intercept(s) , set  

 

 

 

Thus, there are no x-intercepts.

 

Finding a few extra points makes our graph more accurate.

x

  1

  2

  4

-1

-2

  -4

y

  8

  2

 

 8

 2

 

 

Plotting the above, we obtain:

 

 

 

The graph of  

 

 

Example 4.   

 

Set the denominator = 0 to find the values for which the function is undefined.  The domain is all real numbers EXCEPT this value.

 

 is the domain

Choose some very large values for x and some very small values for x to determine end behavior.

 

As  

As  

The vertical asymptote(s) occurs at the value(s) for which the function is undefined.

 

The vertical asymptote is  

The end behavior determines the horizontal asymptote.

The horizontal asymptote is  

 

To find the y-intercept, set  

, which is undefined.  There is no

               y-intercept.

To find the x-intercept(s) , set  

 

 

 

Thus, there are no x-intercepts.

 

Finding a few extra points makes our graph more accurate.

x

1

2

 

 

-1

-2

 

y

-2

  

-16

2

 

16

 

Plotting the above, we obtain:

 

 

 

The graph of  

 

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