4-1 Rational Functions (Presentation)

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4-1 Rational Functions

Unit 4 Rational Algebraic Functions



Concepts and ObjectivesRational Functions (Obj. #12)

Identify and graph horizontal and verticaltranslations of the reciprocal functionDetermine vertical, horizontal, and oblique

Graph rational functions with no common terms



Rational FunctionsA rational function is a function of the form

where p( x ) and q( x ) are polynomials, with q( x ) ≠ 0.

( ) ( )( )

= p x f x q x

The simplest rational function with a variabledenominator is the reciprocal function , defined by

( )= ≠1 , 0 f x x



The Reciprocal FunctionAs x gets closer and closer to 0, the value of f ( x ) gets

larger and larger (or smaller and smaller)

x y

1 1

21

2

0.1 10

0.01 100





Translating, RevisitedRecall from last week:

( )=2 f x x ( )=

2h x x

( )= −2 2 g x x ( ) ( )= −

21 j x x



Translating, RevisitedWe could also write this as

( )=2 f x x ( )=

2h x x

( ) ( )= − 2 g x f x ( ) ( )= − 1 j x h x



Translating, RevisitedSo, looking at translations of the reciprocal function, f :

( ) ( )= = −−

11. 3

3 g x f x

x

shifted 3 units to the right

shifted 2 units to the left, up 1

( ) ( )= + = + ++

12. 1 2 1

2h x f x

x



Rational FunctionsAs we can see from the table,

has range values that are all positive,and like the reciprocal function, get larger and larger, the closer x gets tozero.

( )=2

1 f x

x

The graph looks like this:



Determining AsymptotesVertical Asymptotes

To find vertical asymptotes, set the denominatorequal to 0 and solve for x . If a is a zero of thedenominator, then the line x = a is a vertical

.Example: Find the vertical asymptote(s) of

asymptote is at x = 2

( )=−

3

2

x f x

− =

=

2 0

2 x



Determining AsymptotesHorizontal Asymptotes

If the numerator has lower degree than denominator,then there is a horizontal asymptote at y = 0.If the numerator and denominator have the same

,of the coefficients of the first terms.



Determining AsymptotesExample: Find the horizontal asymptotes of

a) b)

( )+

=−

2

3

16

x f x

x ( )

−=

+

3 4

2 1

x f x

x

a) The numerator has a lower degree (1) than thedenominator (2), so there is a H.A. at y = 0.

b) Since the numerator and denominator have thesame degree, the H.A. is at

=3

2 y



Determining AsymptotesOblique Asymptotes

If the numerator is exactly one degree more than thedenominator, then the function has an oblique(slanted) asymptote .

,and disregard the remainder. Set the rest of thequotient equal to y for the equation of the asymptote.

The graph cannot intersect any vertical asymptote.

There can be at most one other nonvertical asymptote,and the graph can intersect that asymptote.



Determining AsymptotesExample: Find the asymptotes of

( )+

=−

22 53

x f x x

V.A.: O.A.:− =

=3 0

3 x 3 2 0 5

6 18

2 6 23

= +2 6 y x



Graphing a Rational FunctionTo graph a rational function:

1. Find any vertical asymptotes.2. Find any horizontal or oblique asymptotes.3. Find the y -intercept by evaluating f (0 ).

4. Find the -intercepts, if any, by finding the zeros of thenumerator.5. Determine whether the graph will intersect its

nonvertical asymptote by setting the function equal tothe equation of the asymptote.

6. Plot other points, as necessary, and sketch the graph.



Graphing a Rational FunctionExample: Graph

V.A.: H.A.:

( )−

=− −

2

2

6 x

f x x x

− − =2 6 0 x x

( )( )− + =3 2 0 x x y = 0

= − 2, 3 x




y -intercept:

( )−

=− −

2

2

6

x f x

x x

( )−

= =− −

0 2 10

0 0 6 3 f

x -intercept:− =

=2 0

2 x

• •




check H.A.:

( )−

=− −

2

2

6

x f x

x x −

=− −

2

20

6

x x x

− =

crosses the H.A. • •

= 2 x

( )− − −

− = = =+ − −

1 2 3 31 1 1 6 4 4 f

•



Graphing a Rational FunctionExample: Graph ( )

−=

− −2

2

6

x f x

x x

( )− − −

− = =+ −

3 2 53

9 3 6 6 f

• •••

( )− − −

− = = = −+ −4 16 4 6 14 7 f

•( )−

= = =− −

4 2 2 14

16 4 6 6 3 f •



Graphing a Rational FunctionExample: Graph ( )

−=

− −2

2

6

x f x

x x



Homework College Algebra

Page 372: 18-27 ( ×3), 37-48, 65-90 ( ×5)Turn In: 27, 40, 42, 44, 48, 70, 90

4-1 Rational Functions (Presentation)

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