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Pg 1 Lecture Notes by M. Torres Math 114 , College Algebra

Chapter 4(Lecture Notes)

Disclaimer:

For the entire chapter 4 lecture notes,assume that all the graphs extend indefinitely (unless the endpoints are marked).

Section 4.1: Rational Functions and Graphs Definition :

A function f defined by( )

( )( )

p x f x

q x= , where q( x) 0

and p( x) and q( x) are polynomials is called a !"#$%&"'()*&+#$%& .

!"#$%&'#()*+,-#.$'-/*

1( ) f x

x=

*

Domain: ( , 0) ! (0, )Range: ( , 0) ! (0, )

A few things to note about the reciprocal function :! Function is discontinuous at x = 0! Vertical asymptote: x = 0 ( y-axis)! Horizontal asymptore: y = 0 ( x-axis)! Function is odd; graph is symmetric with respect

to the origin.

!(.$'-()*+,-#.$'-/*

2

1( ) f x

x=

*

Domain: ( , 0) ! (0, )Range: (0, )

A few things to note about the rational function :! Function is discontinuous at x = 0! Vertical asymptote: x = 0 ( y-axis)! Horizontal asymptore: y = 0 ( x-axis)! Function is even; graph is symmetric with respect

to y-axis.

Example 1: What is the equation of the vertical asymptote of the graph of 21

4( 2)

y x

= +

? of the

horizontal asymptote?

Remember to read the textbook before attempting to do your homework


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Example 2: Explain how the graph of f can be obtained from the graph of 1

y x

= or 21

y x

= . Draw a

!"#$%& of the graph of f by hand. Give the domain and range.Optional : Generate an accurate depiction of the graph with a graphing calculator.

a.1

2 y x

=

b.2

12

y x

= +

c. 21

3 y x

= +


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d. 21

2( 4)

y x

= +

Example 3: Use division and translations to !"#$%& the graph of 23

x y

x

=

by hand.

Remember to read the textbook before attempting to do your homework.

Section 4.2: More on Graphs of Rational Functions'#$#()*+*+,-.!/)0$1$#!2

Consider a rational function defined by a rational expression that is written in !"#$%&'&$()%.

! To find the *+,-./01'2%3)4&"&$%'5*6067 : o Set the denominator equal to zero and solve for x.

If a is a zero of the denominator, then x = a is the equation of the vertical asymptote.

! Finding 89,.:9;-01'2%3)4&"&$%'586067 : o If the 3#,(## of the numerator and denominator are the 4.56 , then ?C'D"$@@>D>$?&%will give the equation of the horizontal asymptote.

Example :2

2

5 13 2 10

x y

x x

=

+ , since the numerator and denominator are of the same

degree, the equation of the horizontal asymptote is given by y = 5/3 --


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! Finding !"#$%"&'()*+,-./0102,*3!4(45 : (continued )o If the !"#$"" of the numerator is %&''()*+, the degree of the denominator, then the

horizontal asymptote is given by y = 0.

! Finding "6)$789*+,-./0102, (slanted asymptotes)o If the !"#$"" of the numerator is &-+.)%/(0,&(102& than the degree of

denominator, then there may be an oblique asymptote.

Question : How do I find the equation of the oblique asymptote? Answer :

Divide the numerator by the denominator and disregard the remainder. Set the 34567"86( equal to y to get the equation of the asymptote.

Sample :Suppose that when you divide the rational expression you get something like

22 5

5 2 x

x+

the equation of the oblique asymptote is given by y = 2 x + 5(

A few things to note about asymptotes

" The graph .+,,0) intersect any vertical asymptote" There can be only one other ( nonvertical ) asymptote,

and the graph may intersect that asymptote.

.59:$";"8


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!"#$%&'()'*)+($,#-)'*).$/&'0$1)2304/&'0-))

5"$()6"(/&4$1)7-89,/'/"-:)

Let( )

( )( )

p x f x

q x= define a function with rational expression in !"#$%&'&$()%.

If n is the largest positive integer such that ( x a )n is a factor of the *$+"),+-&"( of f ( x), the graphnear the .$(&,/-!'-%0)1&"&$ will behave in the manner illustrated.

If n is even: If n is odd:

If n is the largest positive integer such that ( x c)n is a factor of the +2)$(-&"( of f ( x), the graph

near the x 3,+&$(/$1&will behave in the manner illustrated.If n is even: If n is odd:

Example 1: Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of eachrational function.

a.6

( )9

f x x

=

+

b.2 6

( )4

x f x

x

+=

x x

x

x

x x x x


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c.2 4

( )1

x f x

x

+=

Example 2: !"#$%& a graph of each rational function. Your graph should include all asymptotes andbe comprehensive. You may want to use your calculator as an aid.

a.5

( )3

x f x

x

=

+

b. ( 3)( 5)( )( 1)( 4) x x f x x x

+ =+


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c. 2( ) 9 x

f x x

=

d.

22 3( ) 4

x f x x

+

=

e.2 16

( )4

x f x

x

=

+

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