2.7 graphs of factorable rational functions

Graphs of Rational Functions

http://www.lahc.edu/math/precalculus/math_260a.html

http://www.lahc.edu/math/precalculus/math_260a.html

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)


A rational function is factorable if both P(x) and Q(x) are factorable.

Rational functions are functions of the form



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.




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 thethe same as for polynomials. We study the behaviors and draw pieces of the graphs at important regions, then complete the graphs by connecting them.




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 thethe 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.



Vertical Asymptote Graphs of Rational Functions

Vertical Asymptote The function y = 1/x is not defined at x = 0.


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.


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.


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,


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,


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)


x=0


(1, 1)

(1/2, 2)


x=0


(1, 1)

(1/2, 2)

(1/3, 3)


x=0


(1, 1)

(1/2, 2)

(1/3, 3)

The graph runs along x = 0 but never touches x = 0 as shown.


x=0

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.


(1, 1)

(1/2, 2)

(1/3, 3)

The boundary-line x = 0 is called a vertical asymptote (VA).

x=0


negative x's, y = 1/x are negative


(1, 1)

(1/2, 2)

(1/3, 3)

x=0The boundary-line x = 0 is called a vertical asymptote (VA). For "small"


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)

x=0The boundary-line x = 0 is called a vertical asymptote (VA). For "small"

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)


The boundary-line x = 0 is called a vertical asymptote (VA). For "small"

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)


closer and closer to the x-axis. Hence the x-axis a horizontal asymptote (HA).

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.



However, since 1/x2 is always positive so the graph goes upward along both sides of the asymptotes.


Graph of y = 1/x2


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.


Graph of y = 1/x2




Graph of y = 1/x2

I. The vertical asymptotes take place at where the denominatoris 0 (i.e.. the root of Q(x)).




Graph of y = 1/x2

I. The vertical asymptotes take place at where the denominatoris 0 (i.e.. the root of Q(x)).II. The graph runs along either side of the vertical asymptotes.




Graph of y = 1/x2

I. The vertical asymptotes take place at where the denominatoris 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.




Graph of y = 1/x2

I. The vertical asymptotes take place at where the denominatoris 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.

Graphs of Rational Functions The four cases of graphs along a vertical asymptote:


+

e.g. y = 1/x

The four cases of graphs along a vertical asymptote:


e.g. y = -1/x

+

e.g. y = 1/x

+



e.g. y = 1/x2

e.g. y = -1/x

+

e.g. y = 1/x

++ +



e.g. y = 1/x2

e.g. y = -1/x

e.g. y = -1/x2

+

e.g. y = 1/x

++ +



e.g. y = 1/x2

e.g. y = -1/x

e.g. y = -1/x2

+

e.g. y = 1/x

++ +


Example A: Given the following information of roots, sign-chart and vertical asymptotes, draw the graph.


e.g. y = 1/x2

e.g. y = -1/x

e.g. y = -1/x2

+

e.g. y = 1/x

++ +



++root

VAVA


e.g. y = 1/x2

e.g. y = -1/x

e.g. y = -1/x2

+

e.g. y = 1/x

++ +



++root

VAVA


e.g. y = 1/x2

e.g. y = -1/x

e.g. y = -1/x2

+

e.g. y = 1/x

++ +



++root

VAVA


e.g. y = 1/x2

e.g. y = -1/x

e.g. y = -1/x2

+

e.g. y = 1/x

++ +



++root

VAVA


e.g. y = 1/x2

e.g. y = -1/x

e.g. y = -1/x2

+

e.g. y = 1/x

++ +



++root

VAVA

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),


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 termsBxK + lower degree terms

Specifically, if



R(x) =


Specifically, if

then for x's where | x | is large, the graph of R(x)

resembles (quotient of the leading terms).

AxN BxK



R(x) =


Specifically, if



AxN BxK

The graph may or may not level off horizontally.



R(x) =


Specifically, if



AxN BxK

The graph may or may not level off horizontally. If it does, then we have a horizontal asymptote (HA).

Graphs of Rational Functions We list all the possibilities of horizontal behavior below:

Graphs of Rational Functions We list all the possibilities of horizontal behavior below: Theorem (Horizontal Behavior):


Given that R(x) =


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.

,


Given that R(x) =




I. If N > K,

,


Given that R(x) =




I. If N > K, then the graph of R(x) resembles the polynomial AxN-K/B.

,


Given that R(x) =





, We write this as lim y = ±∞.

x±∞


Given that R(x) =





II. If N = K,


x±∞


Given that R(x) =





II. If N = K, then the graph of R(x) has y = A/B as a horizontal asymptote (HA).


x±∞


Given that R(x) =





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±∞


x±∞


Given that R(x) =






III. If N < K, x±∞


x±∞


Given that R(x) =






III. If N < K, then the graph of R(x) has y = 0 as a horizontal asymptote (HA) because N – K is negative.

x±∞


x±∞


Given that R(x) =






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±∞


x±∞

Graphs of Rational Functions Steps for graphing rational functions R(x) =

P(x)Q(x)

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)


P(x)Q(x)

II. (VA) Find the vertical asymptotes (VA) of R(x) and their orders by solving Q(x) = 0.


P(x)Q(x)


Steps I and II give the signed-chart,


P(x)Q(x)


Steps I and II give the signed-chart, the graphs around the roots (using their orders)


P(x)Q(x)


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).


P(x)Q(x)


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).


P(x)Q(x)


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.


x±∞

P(x)Q(x)


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 .


x±∞

P(x)Q(x)


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)

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


x2 – 4x + 4 x2 – 1

For it's root, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).


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.


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


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


x2 – 4x + 4 x2 – 1


+–x=2

++


x2 – 4x + 4 x2 – 1


++ –x=2

+ +


x2 – 4x + 4 x2 – 1


++ –x=2

+ +


x2 – 4x + 4 x2 – 1


++ –x=2

+ +


x2 – 4x + 4 x2 – 1


++ –x=2

+ +


x2 – 4x + 4 x2 – 1


++ –x=2

+ +


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

+ +


x2 – 4x + 4 x2 – 1



++ –x=2

+ +


x2 – 4x + 4 x2 – 1



++ –x=2

++


x2 – 4x + 4 x2 – 1


= 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

++

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


x2 – 2x – 3 x – 2

Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.


x2 – 2x – 3 x – 2

Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.


x2 – 2x – 3 x – 2


Do the sign-chart.


x2 – 2x – 3 x – 2


x=3

Do the sign-chart.


x2 – 2x – 3 x – 2


x=3

Do the sign-chart.

+–+–


x2 – 2x – 3 x – 2


Do the sign-chart. Construct the middle part of the graph.

x=3+–+–


x2 – 2x – 3 x – 2



x=3+–+–


x2 – 2x – 3 x – 2



x=3+–+–


x2 – 2x – 3 x – 2



x=3+–+–


x2 – 2x – 3 x – 2



x=3+–+–


x2 – 2x – 3 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+–+–



x2 – 2x – 3 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+–+–



x2 – 2x – 3 x – 2



x=3+–+–



x2 – 2x – 3 x – 2



x=3


+–+–

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2.7 graphs of factorable rational functions