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Graphs of Ration= al Functions
Recalled that <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >rational functions are functions of the
form R(x) =3D        = ;   where P(x) and Q(x) are
polynomials.
P(x)
Q(x)
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Graphs of Ration= al Functions
Recalled that <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >rational functions are functions of the
form R(x) =3D        = ;   where P(x) and Q(x) are
polynomials.
P(x)
Q(x)
A rational function is factorable if both P(x) and <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>Q(x) are factorable. 
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Graphs of Ration= al Functions
Recalled that <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >rational functions are functions of the
form R(x) =3D        = ;   where P(x) and Q(x) are
polynomials.
P(x)
Q(x)
A rational function is factorable if both P(x) and <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>Q(x) are factorable. 
In this section we study the graphs of reduced = factorable rational functions.   
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Graphs of Ration= al Functions
Recalled that <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >rational functions are functions of the
form R(x) =3D        = ;   where P(x) and Q(x) are
polynomials.
P(x)
Q(x)
A rational function is factorable if both P(x) and <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>Q(x) are factorable. 
In this section we study the graphs of reduced = factorable rational functions.   
The main principle of graphing these functions is the
the same as polynomial= s.
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Graphs of Ration= al Functions
Recalled that <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >rational functions are functions of the
form R(x) =3D        = ;   where P(x) and Q(x) are
polynomials.
P(x)
Q(x)
A rational function is factorable if both P(x) and <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>Q(x) are factorable. 
In this section we study the graphs of reduced = factorable rational functions.   
The main principle of graphing these functions is the
the same as polynomial= s. We analyze the behaviors and draw pieces the graphs at important locati= ons, then complete the graphs by connecting them.  
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Graphs of Ration= al Functions
Recalled that <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >rational functions are functions of the
form R(x) =3D        = ;   where P(x) and Q(x) are
polynomials.
P(x)
Q(x)
A rational function is factorable if both P(x) and <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>Q(x) are factorable. 
In this section we study the graphs of reduced = factorable rational functions.   
The main principle of graphing these functions is the
the same as polynomial= s. We analyze the behaviors and draw pieces the graphs at important locati= ons, then complete the graphs by connecting them.  
However the behaviors of rational functions are more complicat= ed due to the presence of the denominators.  
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Vertical Asymptote
The function y =3D 1/x is not defined at x =3D 0. So the = graph is = not a continuous curve, it breaks at x =3D 0.       
For small positive x's, y =3D 1/x is large.
Graphs of Ration= al Functions
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Vertical Asymptote
The function y =3D 1/x is not defined at x =3D 0. So the = graph is = not a continuous curve, it breaks at x =3D 0.       
For small positive x's, y =3D 1/x is large.
The closer the x is to 0, the smaller x is, <= /span>
Graphs of Ration= al Functions
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Vertical Asymptote
The function y =3D 1/x is not defined at x =3D 0. So the = graph is = not a continuous curve, it breaks at x =3D 0.       
For small positive x's, y =3D 1/x is large.
The closer the x is to 0, the smaller x is, correspondingly the larger y =3D 1/x is,
Graphs of Ration= al Functions
------=_NextPart_01C8347A.B537DA10 Content-Location: file:///C:/4D764A33/GraphsofFactorableRationalFunctions_files/slide0339.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Sign Charts and Inequlities
Vertical Asymptote
The function y =3D 1/x is not defined at x =3D 0. So the = graph is = not a continuous curve, it breaks at x =3D 0.       
For small positive x's, y =3D 1/x is large.
The closer the x is to 0, the smaller x is, correspondingly the larger y =3D 1/x is, hence <= span style=3D'position:absolute;top:53.0%;left:4.3%;width:61.79%;height:6.0%'><= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>the highe= r the point (x, 1/x) is.
Graphs of Ration= al Functions
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Vertical Asymptote
The function y =3D 1/x is not defined at x =3D 0. So the = graph is = not a continuous curve, it breaks at x =3D 0.       
For small positive x's, y =3D 1/x is large.
The closer the x is to 0, the smaller x is, correspondingly the larger y =3D 1/x is, hence <= span style=3D'position:absolute;top:53.0%;left:4.3%;width:61.79%;height:6.0%'><= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>the highe= r the point (x, 1/x) is.
(1, 1)
Graphs of Ration= al Functions
x=3D0
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Vertical Asymptote
The function y =3D 1/x is not defined at x =3D 0. So the = graph is = not a continuous curve, it breaks at x =3D 0.       
For small positive x's, y =3D 1/x is large.
The closer the x is to 0, the smaller x is, correspondingly the larger y =3D 1/x is, hence <= span style=3D'position:absolute;top:53.0%;left:4.3%;width:61.79%;height:6.0%'><= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>the highe= r the point (x, 1/x) is.
(1, 1)
(1/2, 2)
Graphs of Ration= al Functions
x=3D0
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Vertical Asymptote
The function y =3D 1/x is not defined at x =3D 0. So the = graph is = not a continuous curve, it breaks at x =3D 0.       
For small positive x's, y =3D 1/x is large.
The closer the x is to 0, the smaller x is, correspondingly the larger y =3D 1/x is, hence <= span style=3D'position:absolute;top:53.0%;left:4.3%;width:61.79%;height:6.0%'><= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>the highe= r the point (x, 1/x) is.
(1, 1)
(1/2, 2)
(1/3, 3)
Graphs of Ration= al Functions
x=3D0
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Vertical Asymptote
The function y =3D 1/x is not defined at x =3D 0. So the = graph is = not a continuous curve, it breaks at x =3D 0.       
For small positive x's, y =3D 1/x is large.
The closer the x is to 0, the smaller x is, correspondingly the larger y =3D 1/x is, hence <= span style=3D'position:absolute;top:53.0%;left:4.3%;width:61.79%;height:6.0%'><= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>the highe= r the point (x, 1/x) is.
(1, 1)
(1/2, 2)
(1/3, 3)
The graph runs along x =3D 0 but never touches x =3D 0 = as shown.
Graphs of Ration= al Functions
x=3D0
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Vertical Asymptote
The function y =3D 1/x is not defined at x =3D 0. So the = graph is = not a continuous curve, it breaks at x =3D 0.       
For small positive x's, y =3D 1/x is large.
The closer the x is to 0, the smaller x is, correspondingly the larger y =3D 1/x is, hence <= span style=3D'position:absolute;top:53.0%;left:4.3%;width:61.79%;height:6.0%'><= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>the highe= r the point (x, 1/x) is.
The graph runs along x =3D 0 but never touches x =3D 0 = as shown.
Graphs of Ration= al Functions
(1, 1)
(1/2, 2)
(1/3, 3)
The boundary-line x =3D 0 is called a vertical<= /i> <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >asymptote (VA).
x=3D0
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Vertical Asymptote
The function y =3D 1/x is not defined at x =3D 0. So the = graph is = not a continuous curve, it breaks at x =3D 0.       
For small positive x's, y =3D 1/x is large.
The closer the x is to 0, the smaller x is, correspondingly the larger y =3D 1/x is, hence <= span style=3D'position:absolute;top:53.0%;left:4.3%;width:61.79%;height:6.0%'><= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>the highe= r the point (x, 1/x) is.
The graph runs along x =3D 0 but never touches x =3D 0 = as shown.
negative = x's, y =3D 1/x are negative
Graphs of Ration= al Functions
(1, 1)
(1/2, 2)
(1/3, 3)
x=3D0
The boundary-line x =3D 0 is called a vertical<= /i> <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >asymptote (VA). For "small"
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Vertical Asymptote
The function y =3D 1/x is not defined at x =3D 0. So the = graph is = not a continuous curve, it breaks at x =3D 0.       
For small positive x's, y =3D 1/x is large.
The closer the x is to 0, the smaller x is, correspondingly the larger y =3D 1/x is, hence <= span style=3D'position:absolute;top:53.0%;left:4.3%;width:61.79%;height:6.0%'><= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>the highe= r the point (x, 1/x) is.
The graph runs along x =3D 0 but never touches x =3D 0 = as shown.
negative x's, y =3D 1/x are negative so the corresponding = graph goes downward along the asymptote as shown.&nb= sp;
Graphs of Ration= al Functions
(1, 1)
(1/2, 2)
(1/3, 3)
x=3D0
The boundary-line x =3D 0 is called a vertical<= /i> <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >asymptote (VA). For "small"
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Vertical Asymptote
x=3D0
The function y =3D 1/x is not defined at x =3D 0. So the = graph is = not a continuous curve, it breaks at x =3D 0.       
For small positive x's, y =3D 1/x is large.
The closer the x is to 0, the smaller x is, correspondingly the larger y =3D 1/x is, hence <= span style=3D'position:absolute;top:53.0%;left:4.3%;width:61.79%;height:6.0%'><= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>the highe= r the point (x, 1/x) is.
(1, 1)
(1/2, 2)
(1/3, 3)
The graph runs along x =3D 0 but never touches x =3D 0 = as shown.
negative x's, y =3D 1/x are negative so the corresponding = graph goes downward along the asymptote as shown.&nb= sp;
(-1, -1)
(-1/2, -2)
(-1/3, -3)
Graphs of Ration= al Functions
The boundary-line x =3D 0 is called a vertical<= /i> <= span lang=3DEN-US style=3D'font-size:156%;color:red;mso-fareast-language:ZH-CN'= >asymptote (VA). For "small"
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Likewise, y =3D 1/x2 has x =3D= 0 as vertical asymptote. However, since 1/x2 = is always positive so the graph goes upward along both= sides of the asymptotes.
We list the following facts about vertical asymptotes of= a reduced rational function. 
Graphs of Ration= al Functions
Graph o= f y =3D 1/x2<= span lang=3DEN-US style=3D'font-size:89%;mso-fareast-language:ZH-CN'>   
I. The vertical asympt= otes take place at where the denominator
is 0 (i.e.. the root of Q(x)).
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Likewise, y =3D 1/x2 has x =3D= 0 as vertical asymptote. However, since 1/x2 = is always positive so the graph goes upward along both= sides of the asymptotes.
We list the following facts about vertical asymptotes of= a reduced rational function. 
Graphs of Ration= al Functions
Graph o= f y =3D 1/x2<= span lang=3DEN-US style=3D'font-size:89%;mso-fareast-language:ZH-CN'>   
I. The vertical asympt= otes take place at where the denominator
is 0 (i.e.. the root of Q(x)).
II. The graph runs along either side of the vertical asymptotes.
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Likewise, y =3D 1/x2 has x =3D= 0 as vertical asymptote. However, since 1/x2 = is always positive so the graph goes upward along both= sides of the asymptotes.
We list the following facts about vertical asymptotes of= a reduced rational function. 
Graphs of Ration= al Functions
Graph o= f y =3D 1/x2<= span lang=3DEN-US style=3D'font-size:89%;mso-fareast-language:ZH-CN'>   
I. The vertical asympt= otes take place at where the denominator
is 0 (i.e.. the root of Q(x)).
II. The graph runs along either side of the vertical asymptotes.
Whether the graph goes upward or downward along the asymptote ma= y be determined using the sing-chart.
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Likewise, y =3D 1/x2 has x =3D= 0 as vertical asymptote. However, since 1/x2 = is always positive so the graph goes upward along both= sides of the asymptotes.
We list the following facts about vertical asymptotes of= a reduced rational function. 
Graphs of Ration= al Functions
Graph o= f y =3D 1/x2<= span lang=3DEN-US style=3D'font-size:89%;mso-fareast-language:ZH-CN'>   
I. The vertical asympt= otes take place at where the denominator
is 0 (i.e.. the root of Q(x)).
II. The graph runs along either side of the vertical asymptotes.
Whether the graph goes upward or downward along the asymptote ma= y be determined using the sing-chart. There are four different cases.
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Graphs of Ration= al Functions
Horizontal Asymptotes
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Graphs of Ration= al Functions
Horizontal Asymptotes
For x's where | x | is large (i.e.. x is to the far right or = far left = of the x-axis),
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Graphs of Ration= al Functions
Horizontal Asymptotes
For x's where | x | is large (i.e.. x is to the far right or far left = of the x-axis), the graph of a rational function <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>resembles= the quotient of the leading terms of the numerator= and the denominator.  =
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Graphs of Ration= al Functions
Horizontal Asymptotes
For x's where | x | is large (i.e.. x is to the far right or far left = of the x-axis), the graph of a rational function <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>resembles= the quotient of the leading terms of the numerator= and the denominator.  =
R(x) =3D<= span style=3D'mso-spacerun:yes'>        = ; 
Ax + lower degree terms
Bx + lower degree terms
Specifica= lly, if        
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Graphs of Ration= al Functions
Horizontal Asymptotes
For x's where | x | is large (i.e.. x is to the far right or far left = of the x-axis), the graph of a rational function <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>resembles= the quotient of the leading terms of the numerator= and the denominator.  =
R(x) =3D<= span style=3D'mso-spacerun:yes'>        = ; 
Ax + lower degree terms
Bx + lower degree terms
Specifica= lly, if        
then for x's where | x= | is large, the graph of R(x)
resembles        = ;  (quotient of the leading terms).        = ; 
Ax 
Bx
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Graphs of Ration= al Functions
Horizontal Asymptotes
For x's where | x | is large (i.e.. x is to the far right or far left = of the x-axis), the graph of a rational function <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>resembles= the quotient of the leading terms of the numerator= and the denominator.  =
R(x) =3D<= span style=3D'mso-spacerun:yes'>        = ; 
Ax + lower degree terms
Bx + lower degree terms
Specifica= lly, if        
then for x's where | x= | is large, the graph of R(x)
resembles        = ;  (quotient of the leading terms).        = ; 
Ax 
Bx
The graph= may or ma= y not level off horizontally.
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Graphs of Ration= al Functions
Horizontal Asymptotes
For x's where | x | is large (i.e.. x is to the far right or far left = of the x-axis), the graph of a rational function <= span lang=3DEN-US style=3D'font-size:156%;mso-fareast-language:ZH-CN'>resembles= the quotient of the leading terms of the numerator= and the denominator.  =
R(x) =3D<= span style=3D'mso-spacerun:yes'>        = ; 
Ax + lower degree terms
Bx + lower degree terms
Specifica= lly, if        
then for x's where | x= | is large, the graph of R(x)
resembles        = ;  (quotient of the leading terms).        = ; 
Ax 
Bx
The graph may or ma= y not level off horizontally.
If it does, then we ha= ve a horizontal asymptote (HA). 
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Graphs of Ration= al Functions
We list a= ll the possibilities of horizontal behavior below: 
Given tha= t R(x) =3D        = ; 
Ax + lower degree terms
Bx + lower degree terms
Theorem (Horizontal Behavior):       
the graph of R(x) as x goes to the far right (x à ∞) and far left (xà -= ∞) behaves similar to Ax<= /span>N/BxK =3D AxN-K/B.
,        = ; 
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Graphs of Ration= al Functions
We list a= ll the possibilities of horizontal behavior below: 
Given tha= t R(x) =3D        = ; 
Ax + lower degree terms
Bx + lower degree terms
Theorem (Horizontal Behavior):       
the graph of R(x) as x goes to the far right (x à ∞) and far left (xà -= ∞) behaves similar to Ax<= /span>N/BxK =3D AxN-K/B.
I. If N &= gt; K,
,        = ; 
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Graphs of Ration= al Functions
We list a= ll the possibilities of horizontal behavior below: 
Given tha= t R(x) =3D        = ; 
Ax + lower degree terms
Bx + lower degree terms
Theorem (Horizontal Behavior):       
the graph of R(x) as x goes to the far right (x à ∞) and far left (xà -= ∞) behaves similar to Ax<= /span>N/BxK =3D AxN-K/B.
I. If N > K, then the graph of R(x) resemble the = polynomia= l AxN-K/B.
,        = ; 
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Graphs of Ration= al Functions
We list a= ll the possibilities of horizontal behavior below: 
Given tha= t R(x) =3D        = ; 
Ax + lower degree terms
Bx + lower degree terms
Theorem (Horizontal Behavior):       
the graph of R(x) as x goes to the far right (x à ∞) and far left (xà -= ∞) behaves similar to Ax<= /span>N/BxK =3D AxN-K/B.
I. If N > K, then the graph of R(x) resemble the = polynomia= l AxN-K/B.
,        = ; 
We write = this as lim  y =3D ±∞.
x<= b>à±∞
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Graphs of Ration= al Functions
We list a= ll the possibilities of horizontal behavior below: 
Given tha= t R(x) =3D        = ; 
Ax + lower degree terms
Bx + lower degree terms
Theorem (Horizontal Behavior):       
the graph of R(x) as x goes to the far right (x à ∞) and far left (xà -= ∞) behaves similar to Ax<= /span>N/BxK =3D AxN-K/B.
I. If N > K, then the graph of R(x) resemble the = polynomia= l AxN-K/B.
II. If N = =3D K,
,        = ; 
We write = this as lim  y =3D ±∞.
x<= b>à±∞
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Graphs of Ration= al Functions
We list a= ll the possibilities of horizontal behavior below: 
Given tha= t R(x) =3D        = ; 
Ax + lower degree terms
Bx + lower degree terms
Theorem (Horizontal Behavior):       
the graph of R(x) as x goes to the far right (x à ∞) and far left (xà -= ∞) behaves similar to Ax<= /span>N/BxK =3D AxN-K/B.
I. If N > K, then the graph of R(x) resemble the = polynomia= l AxN-K/B.
II. If N =3D K, then t= he graph of R(x) has y =3D A/B as a
horizontal asymptote (= HA).
,        = ; 
We write = this as lim  y =3D ±∞.
x<= b>à±∞
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Graphs of Ration= al Functions
We list a= ll the possibilities of horizontal behavior below: 
Given tha= t R(x) =3D        = ; 
Ax + lower degree terms
Bx + lower degree terms
Theorem (Horizontal Behavior):       
the graph of R(x) as x goes to the far right (x à ∞) and far left (xà -= ∞) behaves similar to Ax<= /span>N/BxK =3D AxN-K/B.
I. If N > K, then the graph of R(x) resemble the = polynomia= l AxN-K/B.
II. If N =3D K, then t= he graph of R(x) has y =3D A/B as a
horizontal asymptote (= HA). It is noted as lim  y =3D A/B= .
x<= b>à±∞
,        = ; 
We write = this as lim  y =3D ±∞.
x<= b>à±∞
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Graphs of Ration= al Functions
We list a= ll the possibilities of horizontal behavior below: 
Given tha= t R(x) =3D        = ; 
Ax + lower degree terms
Bx + lower degree terms
Theorem (Horizontal Behavior):       
the graph of R(x) as x goes to the far right (x à ∞) and far left (xà -= ∞) behaves similar to Ax<= /span>N/BxK =3D AxN-K/B.
I. If N > K, then the graph of R(x) resemble the = polynomia= l AxN-K/B.
II. If N =3D K, then t= he graph of R(x) has y =3D A/B as a
horizontal asymptote (= HA). It is noted as lim  y =3D A/B= .
III. If N= < K,
x<= b>à±∞
,        = ; 
We write = this as lim  y =3D ±∞.
x<= b>à±∞
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Graphs of Ration= al Functions
We list a= ll the possibilities of horizontal behavior below: 
Given tha= t R(x) =3D        = ; 
Ax + lower degree terms
Bx + lower degree terms
Theorem (Horizontal Behavior):       
the graph of R(x) as x goes to the far right (x à ∞) and far left (xà -= ∞) behaves similar to Ax<= /span>N/BxK =3D AxN-K/B.
I. If N > K, then the graph of R(x) resemble the = polynomia= l AxN-K/B.
II. If N =3D K, then t= he graph of R(x) has y =3D A/B as a
horizontal asymptote (= HA). It is noted as lim  y =3D A/B= .
III. If N < K, then the graph of R(x) has y =3D 0 as a horizontal asymptote (HA) because N – K is negative.
x<= b>à±∞
,        = ; 
We write = this as lim  y =3D ±∞.
x<= b>à±∞
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Graphs of Ration= al Functions
We list a= ll the possibilities of horizontal behavior below: 
Given tha= t R(x) =3D        = ; 
Ax + lower degree terms
Bx + lower degree terms
Theorem (Horizontal Behavior):       
the graph of R(x) as x goes to the far right (x à ∞) and far left (xà -= ∞) behaves similar to Ax<= /span>N/BxK =3D AxN-K/B.
I. If N > K, then the graph of R(x) resemble the = polynomia= l AxN-K/B.
II. If N =3D K, then t= he graph of R(x) has y =3D A/B as a
horizontal asymptote (= HA). It is noted as lim  y =3D A/B= .
III. If N < K, then= the graph of R(x) has y =3D 0 as a horizontal asymptote (= HA) because N – K is negative.
It is noted as lim y = =3D 0.
x<= b>à±∞
x<= b>à±∞
,        = ; 
We write = this as lim  y =3D ±∞.
x<= b>à±∞
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Graphs of Ration= al Functions
Steps = for graphing rational functions R(x) =3D        
P(x)
Q(x)
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Graphs of Ration= al Functions
Steps = for graphing rational functions R(x) =3D        
I. (Roots) As for graphing polynomials, find the roots of R(x) a= nd their orders by solving R(x) =3D 0.
P(x)
Q(x)
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Graphs of Ration= al Functions
Steps = for graphing rational functions R(x) =3D        
I. (Roots) As for graphing polynomials, find the roots of R(x) a= nd their orders by solving R(x) =3D 0.
P(x)
Q(x)
II. (VA) Find the vertical asymptotes (VA) of R(x) and their ord= ers by solving Q(x) =3D 0.
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