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Graphs of Ration= al 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)
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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
Graphs of Ration= al Functions
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Vertical Asymptote
The funct= ion y =3D 1/x is not defined at x =3D 0.
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.       
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.
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
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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.
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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Graph of = y =3D 1/x   
x=3D0
As x gets larger and larger, the corresponding y =3D 1/x = become sm= aller and smaller.  This means the = graph = gets clos= er and closer to the x-axis as it goes to      
further and further to= the right and to the left.  To t= he right, because y =3D 1/x is positive, the graph stays above the =
x-axis. To the left, y= =3D 1/x is negative so the graph stays below the x-axis.  As the graph = goes furt= her to the left. It gets
(1, 1)
(2, 1/2)
(3, 1/3)
(-1, -1)
(-2, -1/2)
(-3, -1/3)
Graphs of Ration= al Functions
closer and closer to t= he x-axis.
Hence the x-axis a horizontal asymptote (HA). 
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Likewise,= y =3D 1/x2 has x =3D 0 as vertical asymptote.
Graphs of Ration= al Functions
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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.
Graphs of Ration= al Functions
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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.
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'>   
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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'>   
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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
The four = cases of graphs along a vertical asymptote:
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Graphs of Ration= al Functions
+
e.g. y = =3D 1/x    <= /div>
The four = cases of graphs along a vertical asymptote:
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Graphs of Ration= al Functions
e.g. y = =3D -1/x    =
+
e.g. y = =3D 1/x    <= /div>
+
The four = cases of graphs along a vertical asymptote:
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Graphs of Ration= al Functions
e.g. y = =3D 1/x2   
e.g. y = =3D -1/x    =
+
e.g. y = =3D 1/x    <= /div>
+
+
+
The four = cases of graphs along a vertical asymptote:
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Graphs of Ration= al Functions
e.g. y = =3D 1/x2   
e.g. y = =3D -1/x    =
e.g. y = =3D -1/x2   
+
e.g. y = =3D 1/x    <= /div>
+
+
+
The four = cases of graphs along a vertical asymptote:
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Graphs of Ration= al Functions
e.g. y = =3D 1/x2   
e.g. y = =3D -1/x    =
e.g. y = =3D -1/x2   
+
e.g. y = =3D 1/x    <= /div>
+
+
+
The four = cases of graphs along a vertical asymptote:
Example A:  Given the = <= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>following information of roots, sign-chart and = vertical asymptotes, draw the graph.
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Graphs of Ration= al Functions
e.g. y = =3D 1/x2   
e.g. y = =3D -1/x    =
e.g. y = =3D -1/x2   
+
e.g. y = =3D 1/x    <= /div>
+
+
+
The four = cases of graphs along a vertical asymptote:
Example A:  Given the = <= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>following information of roots, sign-chart and = vertical asymptotes, draw the graph.
+
+
root
VA
VA
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Graphs of Ration= al Functions
e.g. y = =3D 1/x2   
e.g. y = =3D -1/x    =
e.g. y = =3D -1/x2   
+
e.g. y = =3D 1/x    <= /div>
+
+
+
The four = cases of graphs along a vertical asymptote:
Example A:  Given the = <= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>following information of roots, sign-chart and = vertical asymptotes, draw the graph.
+
+
root
VA
VA
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Graphs of Ration= al Functions
e.g. y = =3D 1/x2   
e.g. y = =3D -1/x    =
e.g. y = =3D -1/x2   
+
e.g. y = =3D 1/x    <= /div>
+
+
+
The four = cases of graphs along a vertical asymptote:
Example A:  Given the = <= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>following information of roots, sign-chart and = vertical asymptotes, draw the graph.
+
+
root
VA
VA
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Graphs of Ration= al Functions
e.g. y = =3D 1/x2   
e.g. y = =3D -1/x    =
e.g. y = =3D -1/x2   
+
e.g. y = =3D 1/x    <= /div>
+
+
+
The four = cases of graphs along a vertical asymptote:
Example A:  Given the = <= span lang=3DEN-US style=3D'font-size:156%;color:#003300;mso-fareast-language:ZH= -CN'>following information of roots, sign-chart and = vertical asymptotes, draw the graph.
+
+
root
VA
VA
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