Welcomeeeee to CHAPTER 2, SECTION 7.  The section about graphs of rational functions.  Neat. 

 

 

 

For starters, here are the basic guidelines for  Graphing Rational Functions.

 

 

 1) Find the y-intercepts.

             (Put 0 in for X)

    2) Find the x-intercepts.

               (Set the numerator = 0)

     3) Find the Vertical Asymptotes (V.A.) by setting the denominator = 0.

     4) Find the Horizontal Asymptote (H.A).

    5) Plot at least three points on your graph.

     6) Draw your curves.                                         

 

Let's apply theses rules, shall we?                         

We shall..

 

Example 1                                                                                                                                                                                                                                                                                                                                                                   

 

 

 

f (x) = 3x-1

             x

 

We need to find the following to help us graph the equation above:

 

X-Intercept: 1/3 (3x-1=0, 3x=1, x=1/3)

Y-Intercept: None

V.A.: x=0

H.A.: y=3

 

Points to Plot:

 

X	Y
1	2
2	5/2
-1	4
-2	7/2

So, the graph should look like this:                                                       

 

                                                              

 

Example 2                                                                                                                                                                                                                                                                                                                                                                      

 

 

f (x) =   3x  

            x+4

 

 

 x-int: 0 (3x=0, x=0/3)             

 y-int: 0 ((3(0)/(0+4)= 0/4) 

 V.A.: -4 (x+4=0, x=-4)       

 H.A.: 3                                                                                                                                                                                                                                                                                                                                                                                   

 

 

Points to Plot:

 

X	Y
-6	9
-5	15
-2	-3
1	3/5

 

 

 

 And the graph looks like this...

          

 

 

 

 

 

 

Oh, but wait!       There are alsoSLANT ASYPMTOTES!   (check for S.A.'s if there is no horizontal aymptote)

 

 

 

 

 

 A Slant Asymptote   only occurs when the    degree of the numerator > the degree of the denomator by one   You can find this using division, such as:

 

 

f(x) = (  - x - 2)

                   x - 1

 

 

 

 - x - 2 = 0

( x + 1 ) ( x - 2 )

 

 

 y-intercept: y=2

 x-intercept: 2, -1

 V.A.: x=1

 H.A.: None.

 S.A.: work shown below.        

 

1| 1  -1  -2   

          1    0         

    1   0   -2            

 

 

 

 

Since the original equation was quadratic this means the answer of your division wil be linear.

 This means that the answer is y=x+0, the slant asymptote is y=x.

 

*remember:

-You can't have a slant asymptote and a horizontal asymptote on the same equation/problem.

-Also after you use division to find the equation for the S.A. throw out any remainder you find.

 

 

 

 

 

 

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