1.
Graphs of Rational Functions
http://www.lahc.edu/math/precalculus/math_260a.html
2.
Graphs of Rational Functions
Rational functions are functions of the form
R(x) = where P(x) and Q(x) are polynomials.
P(x)
Q(x)
3.
Graphs of Rational Functions
A rational function is factorable if both P(x) and Q(x)
are factorable.
Rational functions are functions of the form
R(x) = where P(x) and Q(x) are polynomials.
P(x)
Q(x)
4.
Graphs of Rational Functions
A rational function is factorable if both P(x) and Q(x)
are factorable.
Unless otherwise stated, the rational functions in this
section are assumed to be reduced factorable rational
functions.
Rational functions are functions of the form
R(x) = where P(x) and Q(x) are polynomials.
P(x)
Q(x)
5.
Graphs of Rational Functions
A rational function is factorable if both P(x) and Q(x)
are factorable.
Unless otherwise stated, the rational functions in this
section are assumed to be reduced factorable rational
functions.
The principles of graphing rational functions are the
the same as for polynomials. We study the behaviors
and draw pieces of the graphs at important regions,
then complete the graphs by connecting them.
Rational functions are functions of the form
R(x) = where P(x) and Q(x) are polynomials.
P(x)
Q(x)
6.
Graphs of Rational Functions
A rational function is factorable if both P(x) and Q(x)
are factorable.
Unless otherwise stated, the rational functions in this
section are assumed to be reduced factorable rational
functions.
The principles of graphing rational functions are the
the same as for polynomials. We study the behaviors
and draw pieces of the graphs at important regions,
then complete the graphs by connecting them.
However, the behaviors of rational functions are more
complicated due to the presence of the denominators.
Rational functions are functions of the form
R(x) = where P(x) and Q(x) are polynomials.
P(x)
Q(x)
7.
Vertical Asymptote
Graphs of Rational Functions
8.
Vertical Asymptote
The function y = 1/x is not defined at x = 0.
Graphs of Rational Functions
9.
Vertical Asymptote
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
Graphs of Rational Functions
10.
Vertical Asymptote
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
For small positive x's, y = 1/x is large.
Graphs of Rational Functions
11.
Vertical Asymptote
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
For small positive x's, y = 1/x is large.
The closer the x is to 0, the
smaller x is,
Graphs of Rational Functions
12.
Vertical Asymptote
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
For small positive x's, y = 1/x is large.
The closer the x is to 0, the
smaller x is, correspondingly
the larger y = 1/x is,
Graphs of Rational Functions
13.
Vertical Asymptote
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
For small positive x's, y = 1/x is large.
The closer the x is to 0, the
smaller x is, correspondingly
the larger y = 1/x is, hence
the higher the point (x, 1/x) is.
Graphs of Rational Functions
14.
Vertical Asymptote
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
For small positive x's, y = 1/x is large.
The closer the x is to 0, the
smaller x is, correspondingly
the larger y = 1/x is, hence
the higher the point (x, 1/x) is.
(1, 1)
Graphs of Rational Functions
x=0
15.
Vertical Asymptote
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
For small positive x's, y = 1/x is large.
The closer the x is to 0, the
smaller x is, correspondingly
the larger y = 1/x is, hence
the higher the point (x, 1/x) is.
(1, 1)
(1/2, 2)
Graphs of Rational Functions
x=0
16.
Vertical Asymptote
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
For small positive x's, y = 1/x is large.
The closer the x is to 0, the
smaller x is, correspondingly
the larger y = 1/x is, hence
the higher the point (x, 1/x) is.
(1, 1)
(1/2, 2)
(1/3, 3)
Graphs of Rational Functions
x=0
17.
Vertical Asymptote
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
For small positive x's, y = 1/x is large.
The closer the x is to 0, the
smaller x is, correspondingly
the larger y = 1/x is, hence
the higher the point (x, 1/x) is.
(1, 1)
(1/2, 2)
(1/3, 3)
The graph runs along x = 0 but
never touches x = 0 as shown.
Graphs of Rational Functions
x=0
18.
Vertical Asymptote
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
For small positive x's, y = 1/x is large.
The closer the x is to 0, the
smaller x is, correspondingly
the larger y = 1/x is, hence
the higher the point (x, 1/x) is.
The graph runs along x = 0 but
never touches x = 0 as shown.
Graphs of Rational Functions
(1, 1)
(1/2, 2)
(1/3, 3)
The boundary-line x = 0 is called a
vertical asymptote (VA).
x=0
19.
Vertical Asymptote
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
For small positive x's, y = 1/x is large.
The closer the x is to 0, the
smaller x is, correspondingly
the larger y = 1/x is, hence
the higher the point (x, 1/x) is.
The graph runs along x = 0 but
never touches x = 0 as shown.
negative x's, y = 1/x are negative
Graphs of Rational Functions
(1, 1)
(1/2, 2)
(1/3, 3)
x=0
The boundary-line x = 0 is called a
vertical asymptote (VA). For "small"
20.
Vertical Asymptote
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
For small positive x's, y = 1/x is large.
The closer the x is to 0, the
smaller x is, correspondingly
the larger y = 1/x is, hence
the higher the point (x, 1/x) is.
The graph runs along x = 0 but
never touches x = 0 as shown.
negative x's, y = 1/x are negative so the corresponding
graph goes downward along the asymptote as shown.
Graphs of Rational Functions
(1, 1)
(1/2, 2)
(1/3, 3)
x=0
The boundary-line x = 0 is called a
vertical asymptote (VA). For "small"
21.
Vertical Asymptote
x=0
The function y = 1/x is not defined at x = 0. So the
graph is not a continuous curve, it breaks at x = 0.
For small positive x's, y = 1/x is large.
The closer the x is to 0, the
smaller x is, correspondingly
the larger y = 1/x is, hence
the higher the point (x, 1/x) is.
(1, 1)
(1/2, 2)
(1/3, 3)
The graph runs along x = 0 but
never touches x = 0 as shown.
negative x's, y = 1/x are negative so the corresponding
graph goes downward along the asymptote as shown.
(-1, -1)
(-1/2, -2)
(-1/3, -3)
Graphs of Rational Functions
The boundary-line x = 0 is called a
vertical asymptote (VA). For "small"
22.
Graph of y = 1/x
x=0
As x gets larger and larger, the corresponding y = 1/x
become smaller and smaller. This means the graph
gets closer and closer to the x-axis as it goes to
further and further to the right
and to the left. To the right,
because y = 1/x is positive, the
graph stays above the
x-axis. To the left, y = 1/x is
negative so the graph stays
below the x-axis. As the graph
goes further to the left. It gets
(1, 1)
(2, 1/2) (3, 1/3)
(-1, -1)
(-2, -1/2)
(-3, -1/3)
Graphs of Rational Functions
closer and closer to the x-axis.
Hence the x-axis a horizontal asymptote (HA).
23.
Likewise, y = 1/x2 has x = 0 as vertical asymptote.
Graphs of Rational Functions
24.
Likewise, y = 1/x2 has x = 0 as vertical asymptote.
However, since 1/x2 is always positive so the graph
goes upward along both sides of the asymptotes.
Graphs of Rational Functions
25.
Likewise, y = 1/x2 has x = 0 as vertical asymptote.
However, since 1/x2 is always positive so the graph
goes upward along both sides of the asymptotes.
Graphs of Rational Functions
Graph of y = 1/x2
26.
Likewise, y = 1/x2 has x = 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 Rational Functions
Graph of y = 1/x2
27.
Likewise, y = 1/x2 has x = 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 Rational Functions
Graph of y = 1/x2
I. The vertical asymptotes take
place at where the denominator
is 0 (i.e.. the root of Q(x)).
28.
Likewise, y = 1/x2 has x = 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 Rational Functions
Graph of y = 1/x2
I. The vertical asymptotes 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.
29.
Likewise, y = 1/x2 has x = 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 Rational Functions
Graph of y = 1/x2
I. The vertical asymptotes 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 may be determined using the
sing-chart.
30.
Likewise, y = 1/x2 has x = 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 Rational Functions
Graph of y = 1/x2
I. The vertical asymptotes 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 may be determined using the
sing-chart. There are four different cases.
31.
Graphs of Rational Functions
The four cases of graphs along a vertical asymptote:
32.
Graphs of Rational Functions
+
e.g. y = 1/x
The four cases of graphs along a vertical asymptote:
33.
Graphs of Rational Functions
e.g. y = -1/x
+
e.g. y = 1/x
+
The four cases of graphs along a vertical asymptote:
34.
Graphs of Rational Functions
e.g. y = 1/x2e.g. y = -1/x
+
e.g. y = 1/x
+
+ +
The four cases of graphs along a vertical asymptote:
35.
Graphs of Rational Functions
e.g. y = 1/x2e.g. y = -1/x e.g. y = -1/x2
+
e.g. y = 1/x
+
+ +
The four cases of graphs along a vertical asymptote:
36.
Graphs of Rational Functions
e.g. y = 1/x2e.g. y = -1/x e.g. y = -1/x2
+
e.g. y = 1/x
+
+ +
The four cases of graphs along a vertical asymptote:
Example A: Given the
following information of
roots, sign-chart and
vertical asymptotes,
draw the graph.
37.
Graphs of Rational Functions
e.g. y = 1/x2e.g. y = -1/x e.g. y = -1/x2
+
e.g. y = 1/x
+
+ +
The four cases of graphs along a vertical asymptote:
Example A: Given the
following information of
roots, sign-chart and
vertical asymptotes,
draw the graph.
++
root
VAVA
38.
Graphs of Rational Functions
e.g. y = 1/x2e.g. y = -1/x e.g. y = -1/x2
+
e.g. y = 1/x
+
+ +
The four cases of graphs along a vertical asymptote:
Example A: Given the
following information of
roots, sign-chart and
vertical asymptotes,
draw the graph.
++
root
VAVA
39.
Graphs of Rational Functions
e.g. y = 1/x2e.g. y = -1/x e.g. y = -1/x2
+
e.g. y = 1/x
+
+ +
The four cases of graphs along a vertical asymptote:
Example A: Given the
following information of
roots, sign-chart and
vertical asymptotes,
draw the graph.
++
root
VAVA
40.
Graphs of Rational Functions
e.g. y = 1/x2e.g. y = -1/x e.g. y = -1/x2
+
e.g. y = 1/x
+
+ +
The four cases of graphs along a vertical asymptote:
Example A: Given the
following information of
roots, sign-chart and
vertical asymptotes,
draw the graph.
++
root
VAVA
41.
Graphs of Rational Functions
e.g. y = 1/x2e.g. y = -1/x e.g. y = -1/x2
+
e.g. y = 1/x
+
+ +
The four cases of graphs along a vertical asymptote:
Example A: Given the
following information of
roots, sign-chart and
vertical asymptotes,
draw the graph.
++
root
VAVA
42.
Graphs of Rational Functions
Horizontal Asymptotes
43.
Graphs of Rational Functions
Horizontal Asymptotes
For x's where | x | is large (i.e.. x is to the far right or
far left on the x-axis),
44.
Graphs of Rational Functions
Horizontal Asymptotes
For x's where | x | is large (i.e.. x is to the far right or
far left on the x-axis), the graph of a rational function
resembles the quotient of the leading terms of the
numerator and the denominator.
45.
Graphs of Rational Functions
Horizontal Asymptotes
For x's where | x | is large (i.e.. x is to the far right or
far left on the x-axis), the graph of a rational function
resembles the quotient of the leading terms of the
numerator and the denominator.
R(x) =
AxN + lower degree terms
BxK + lower degree terms
Specifically, if
46.
Graphs of Rational Functions
Horizontal Asymptotes
For x's where | x | is large (i.e.. x is to the far right or
far left on the x-axis), the graph of a rational function
resembles the quotient of the leading terms of the
numerator and the denominator.
R(x) =
AxN + lower degree terms
BxK + lower degree terms
Specifically, if
then for x's where | x | is large, the graph of R(x)
resembles (quotient of the leading terms).AxN
BxK
47.
Graphs of Rational Functions
Horizontal Asymptotes
For x's where | x | is large (i.e.. x is to the far right or
far left on the x-axis), the graph of a rational function
resembles the quotient of the leading terms of the
numerator and the denominator.
R(x) =
AxN + lower degree terms
BxK + lower degree terms
Specifically, if
then for x's where | x | is large, the graph of R(x)
resembles (quotient of the leading terms).AxN
BxK
The graph may or may not level off horizontally.
48.
Graphs of Rational Functions
Horizontal Asymptotes
For x's where | x | is large (i.e.. x is to the far right or
far left on the x-axis), the graph of a rational function
resembles the quotient of the leading terms of the
numerator and the denominator.
R(x) =
AxN + lower degree terms
BxK + lower degree terms
Specifically, if
then for x's where | x | is large, the graph of R(x)
resembles (quotient of the leading terms).AxN
BxK
The graph may or may not level off horizontally.
If it does, then we have a horizontal asymptote (HA).
49.
Graphs of Rational Functions
We list all the possibilities of horizontal behavior below:
50.
Graphs of Rational Functions
We list all the possibilities of horizontal behavior below:
Theorem (Horizontal Behavior):
51.
Graphs of Rational Functions
We list all the possibilities of horizontal behavior below:
Given that R(x) =
AxN + lower degree terms
BxK + lower degree terms
Theorem (Horizontal Behavior):
the graph of R(x) as x goes to the far right (x ∞) and
far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B.
,
52.
Graphs of Rational Functions
We list all the possibilities of horizontal behavior below:
Given that R(x) =
AxN + lower degree terms
BxK + lower degree terms
Theorem (Horizontal Behavior):
the graph of R(x) as x goes to the far right (x ∞) and
far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B.
I. If N > K,
,
53.
Graphs of Rational Functions
We list all the possibilities of horizontal behavior below:
Given that R(x) =
AxN + lower degree terms
BxK + lower degree terms
Theorem (Horizontal Behavior):
the graph of R(x) as x goes to the far right (x ∞) and
far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B.
I. If N > K, then the graph of R(x) resembles the
polynomial AxN-K/B.
,
54.
Graphs of Rational Functions
We list all the possibilities of horizontal behavior below:
Given that R(x) =
AxN + lower degree terms
BxK + lower degree terms
Theorem (Horizontal Behavior):
the graph of R(x) as x goes to the far right (x ∞) and
far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B.
I. If N > K, then the graph of R(x) resembles the
polynomial AxN-K/B.
,
We write this as lim y = ±∞.x±∞
55.
Graphs of Rational Functions
We list all the possibilities of horizontal behavior below:
Given that R(x) =
AxN + lower degree terms
BxK + lower degree terms
Theorem (Horizontal Behavior):
the graph of R(x) as x goes to the far right (x ∞) and
far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B.
I. If N > K, then the graph of R(x) resembles the
polynomial AxN-K/B.
II. If N = K,
,
We write this as lim y = ±∞.x±∞
56.
Graphs of Rational Functions
We list all the possibilities of horizontal behavior below:
Given that R(x) =
AxN + lower degree terms
BxK + lower degree terms
Theorem (Horizontal Behavior):
the graph of R(x) as x goes to the far right (x ∞) and
far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B.
I. If N > K, then the graph of R(x) resembles the
polynomial AxN-K/B.
II. If N = K, then the graph of R(x) has y = A/B as a
horizontal asymptote (HA).
,
We write this as lim y = ±∞.x±∞
57.
Graphs of Rational Functions
We list all the possibilities of horizontal behavior below:
Given that R(x) =
AxN + lower degree terms
BxK + lower degree terms
Theorem (Horizontal Behavior):
the graph of R(x) as x goes to the far right (x ∞) and
far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B.
I. If N > K, then the graph of R(x) resembles the
polynomial AxN-K/B.
II. If N = K, then the graph of R(x) has y = A/B as a
horizontal asymptote (HA). It is noted as lim y = A/B.x±∞
,
We write this as lim y = ±∞.x±∞
58.
Graphs of Rational Functions
We list all the possibilities of horizontal behavior below:
Given that R(x) =
AxN + lower degree terms
BxK + lower degree terms
Theorem (Horizontal Behavior):
the graph of R(x) as x goes to the far right (x ∞) and
far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B.
I. If N > K, then the graph of R(x) resembles the
polynomial AxN-K/B.
II. If N = K, then the graph of R(x) has y = A/B as a
horizontal asymptote (HA). It is noted as lim y = A/B.
III. If N < K,
x±∞
,
We write this as lim y = ±∞.x±∞
59.
Graphs of Rational Functions
We list all the possibilities of horizontal behavior below:
Given that R(x) =
AxN + lower degree terms
BxK + lower degree terms
Theorem (Horizontal Behavior):
the graph of R(x) as x goes to the far right (x ∞) and
far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B.
I. If N > K, then the graph of R(x) resembles the
polynomial AxN-K/B.
II. If N = K, then the graph of R(x) has y = A/B as a
horizontal asymptote (HA). It is noted as lim y = A/B.
III. If N < K, then the graph of R(x) has y = 0 as a
horizontal asymptote (HA) because N – K is negative.
x±∞
,
We write this as lim y = ±∞.x±∞
60.
Graphs of Rational Functions
We list all the possibilities of horizontal behavior below:
Given that R(x) =
AxN + lower degree terms
BxK + lower degree terms
Theorem (Horizontal Behavior):
the graph of R(x) as x goes to the far right (x ∞) and
far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B.
I. If N > K, then the graph of R(x) resembles the
polynomial AxN-K/B.
II. If N = K, then the graph of R(x) has y = A/B as a
horizontal asymptote (HA). It is noted as lim y = A/B.
III. If N < K, then the graph of R(x) has y = 0 as a
horizontal asymptote (HA) because N – K is negative.
It is noted as lim y = 0.
x±∞
x±∞
,
We write this as lim y = ±∞.x±∞
61.
Graphs of Rational Functions
Steps for graphing rational functions R(x) = P(x)
Q(x)
62.
Graphs of Rational Functions
Steps for graphing rational functions R(x) =
I. (Roots) As for graphing polynomials, find the roots
of R(x) and their orders by solving R(x) = 0.
P(x)
Q(x)
63.
Graphs of Rational Functions
Steps for graphing rational functions R(x) =
I. (Roots) As for graphing polynomials, find the roots
of R(x) and their orders by solving R(x) = 0.
P(x)
Q(x)
II. (VA) Find the vertical asymptotes (VA) of R(x) and
their orders by solving Q(x) = 0.
64.
Graphs of Rational Functions
Steps for graphing rational functions R(x) =
I. (Roots) As for graphing polynomials, find the roots
of R(x) and their orders by solving R(x) = 0.
P(x)
Q(x)
II. (VA) Find the vertical asymptotes (VA) of R(x) and
their orders by solving Q(x) = 0.
Steps I and II give the signed-chart,
65.
Graphs of Rational Functions
Steps for graphing rational functions R(x) =
I. (Roots) As for graphing polynomials, find the roots
of R(x) and their orders by solving R(x) = 0.
P(x)
Q(x)
II. (VA) Find the vertical asymptotes (VA) of R(x) and
their orders by solving Q(x) = 0.
Steps I and II give the signed-chart, the graphs
around the roots (using their orders)
66.
Graphs of Rational Functions
Steps for graphing rational functions R(x) =
I. (Roots) As for graphing polynomials, find the roots
of R(x) and their orders by solving R(x) = 0.
P(x)
Q(x)
II. (VA) Find the vertical asymptotes (VA) of R(x) and
their orders by solving Q(x) = 0.
Steps I and II give the signed-chart, the graphs
around the roots (using their orders) and the graph
along the VA (upward or downward).
67.
Graphs of Rational Functions
Steps for graphing rational functions R(x) =
I. (Roots) As for graphing polynomials, find the roots
of R(x) and their orders by solving R(x) = 0.
P(x)
Q(x)
II. (VA) Find the vertical asymptotes (VA) of R(x) and
their orders by solving Q(x) = 0.
Steps I and II give the signed-chart, the graphs
around the roots (using their orders) and the graph
along the VA (upward or downward). From these we
construct the middle portion of the graph (as in
example A).
68.
Graphs of Rational Functions
Steps for graphing rational functions R(x) =
I. (Roots) As for graphing polynomials, find the roots
of R(x) and their orders by solving R(x) = 0.
P(x)
Q(x)
II. (VA) Find the vertical asymptotes (VA) of R(x) and
their orders by solving Q(x) = 0.
Steps I and II give the signed-chart, the graphs
around the roots (using their orders) and the graph
along the VA (upward or downward). From these we
construct the middle portion of the graph (as in
example A). Complete the graph with step III.
69.
Graphs of Rational Functions
Steps for graphing rational functions R(x) =
I. (Roots) As for graphing polynomials, find the roots
of R(x) and their orders by solving R(x) = 0.
x±∞
P(x)
Q(x)
II. (VA) Find the vertical asymptotes (VA) of R(x) and
their orders by solving Q(x) = 0.
Steps I and II give the signed-chart, the graphs
around the roots (using their orders) and the graph
along the VA (upward or downward). From these we
construct the middle portion of the graph (as in
example A). Complete the graph with step III.
III. (HA) Use the above theorem to determine the
behavior of the graph to the far right and left, that is,
as .
70.
Graphs of Rational Functions
Steps for graphing rational functions R(x) =
I. (Roots) As for graphing polynomials, find the roots
of R(x) and their orders by solving R(x) = 0.
x±∞
P(x)
Q(x)
II. (VA) Find the vertical asymptotes (VA) of R(x) and
their orders by solving Q(x) = 0.
Steps I and II give the signed-chart, the graphs
around the roots (using their orders) and the graph
along the VA (upward or downward). From these we
construct the middle portion of the graph (as in
example A). Complete the graph with step III.
III. (HA) Use the above theorem to determine the
behavior of the graph to the far right and left, that is,
as . (HA exists only if deg P < deg Q)
71.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
72.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
73.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
74.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
x=2
75.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
All of them has order 1,
so the sign changes at
each of these values.
x=2
76.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
All of them has order 1,
so the sign changes at
each of these values.
+–
x=2
++
77.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
All of them has order 1,
so the sign changes at
each of these values.
++ –
x=2
+ +
78.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
All of them has order 1,
so the sign changes at
each of these values.
++ –
x=2
+ +
79.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
All of them has order 1,
so the sign changes at
each of these values.
++ –
x=2
+ +
80.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
All of them has order 1,
so the sign changes at
each of these values.
++ –
x=2
+ +
81.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
All of them has order 1,
so the sign changes at
each of these values.
++ –
x=2
+ +
82.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
All of them has order 1,
so the sign changes at
each of these values.
As x±∞, R(x) resembles
x2/x2 = 1, i.e. it has y = 1
as the HA.
++ –
x=2
+ +
83.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
All of them has order 1,
so the sign changes at
each of these values.
As x±∞, R(x) resembles
x2/x2 = 1, i.e. it has y = 1
as the HA.
++ –
x=2
+ +
84.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
All of them has order 1,
so the sign changes at
each of these values.
As x±∞, R(x) resembles
x2/x2 = 1, i.e. it has y = 1
as the HA.
++ –
x=2
++
85.
Graphs of Rational Functions
Example C:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 4x + 4
x2 – 1
For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
All of them has order 1,
so the sign changes at
each of these values.
As x±∞, R(x) resembles
x2/x2 = 1, i.e. it has y = 1
as the HA. Note the
graph stays above the
x-axis to the far left,
and below to the far right.
++ –
x=2
++
86.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
87.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
88.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
89.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
Do the sign-chart.
90.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
x=3
Do the sign-chart.
91.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
x=3
Do the sign-chart.
+–+–
92.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
Do the sign-chart. Construct the
middle part of the graph.
x=3
+–+–
93.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
Do the sign-chart. Construct the
middle part of the graph.
x=3
+–+–
94.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
Do the sign-chart. Construct the
middle part of the graph.
x=3
+–+–
95.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
Do the sign-chart. Construct the
middle part of the graph.
x=3
+–+–
96.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
Do the sign-chart. Construct the
middle part of the graph.
x=3
+–+–
97.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
As x ±∞, the graph of R(x)
resembles the graph of the
quotient of the leading terms
x2/x = x, or y = x.
x=3
+–+–
Do the sign-chart. Construct the
middle part of the graph.
98.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
As x ±∞, the graph of R(x)
resembles the graph of the
quotient of the leading terms
x2/x = x, or y = x.
Hence there is no HA.
x=3
+–+–
Do the sign-chart. Construct the
middle part of the graph.
99.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
As x ±∞, the graph of R(x)
resembles the graph of the
quotient of the leading terms
x2/x = x, or y = x.
Hence there is no HA.
x=3
+–+–
Do the sign-chart. Construct the
middle part of the graph.
100.
Graphs of Rational Functions
Example D:
Find the roots, VA and HA, if any, of R(x) =
Draw the sign-chart and sketch graph.
x2 – 2x – 3
x – 2
Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0
so x = -1, 3 are the roots of order 1.
For VA, set x – 2 = 0, i.e.. x = 2.
As x ±∞, the graph of R(x)
resembles the graph of the
quotient of the leading terms
x2/x = x, or y = x.
Hence there is no HA.
x=3
Do the sign-chart. Construct the
middle part of the graph.
+–+–
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