obj. 9 rational functions (presentation)

7/31/2019 Obj. 9 Rational Functions (Presentation)

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Obj. 9 Rational Functions

Unit 3 Rational Functions


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Concepts and ObjectivesRational Functions (Obj. #9)

Identify and graph horizontal and verticaltranslations of the reciprocal functionDetermine vertical, horizontal, and obliqueasymptotes of rational functionsGraph rational functions with no common terms


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Rational FunctionsA rational function is a function of the form

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

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

( ) ( )( )

= p x f x q x

( )= 1 , 0 f x x x


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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

212

0.1 10

0.01 100


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The Reciprocal FunctionThe graph of f ( x ) will never intersect the vertical line

x = 0, which is called a vertical asymptote .As | x | gets larger and larger, the values of f ( x ) get closerand closer to 0. The line y = 0 is called a horizontal asymptote .


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Translating FunctionsCompare the graphs of the following functions:

( )= 2 f x x

( )= 2 2 g x x

( )= 2h x x

( ) ( )= 21 j x x


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Translating FunctionsWe could also write this as

( )= 2 f x x

( ) ( )= 2 g x f x

( )= 2h x x

( ) ( )= 1 j x h x


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

shifted 3 units to the right

shifted 2 units to the left, up 1

( ) ( )= =

11. 3

3 g x f x

x

( ) ( )= + = + ++

12. 1 2 1

2

h x f x

x


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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.

The graph looks like this:

( )= 21

f x x


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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 verticalasymptote.Example: Find the vertical asymptote(s) of

asymptote is at x = 2

( )=

32

x f x

x

=

=

2 0

2

x

x


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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 samedegree, then the horizontal asymptote is at the ratioof the coefficients of the first terms.


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Determining AsymptotesExample: Find the horizontal asymptotes of

a) b)

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

( )+

=2

316

x f x

x ( )

=

+

3 42 1 x

f x x

=32

y


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Determining AsymptotesExample: Find the asymptotes of

V.A.: O.A.:

( )+

=

22 53

x f x x

=

=3 0

3 x

x 3 2 0 5

6 18

2 6 23

= +2 6 y x


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Graphing a Rational FunctionTo graph a rational function:

1. Make sure the function is written in lowest terms.2. Find any vertical, horizontal, or oblique asymptotes.3. Find the y -intercept by evaluating f (0 ).

4. Find the x -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.( )= x b


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Graphing a Rational FunctionExample: Graph

y -intercept:

x -intercept:

( )

= 2

26

x f x

x x

( )

= =

0 2 10

0 0 6 3 f

=

=2 0

2 x

x


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check H.A.:

crosses the H.A.

( )

= 2

26

x f x

x x

=

2

20

6 x

x x =2 0 x

= 2 x

( )

= = =+

1 2 3 31 1 1 6 4 4 f


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Graphing a Rational FunctionExample: Graph ( )

=

2

26

x f x

x x

( )

= =+

3 2 53

9 3 6 6 f

( )

= = = +

4 2 6 34 16 4 6 14 7 f

( )

= = =

4 2 2 14

16 4 6 6 3 f


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=

2

26

x f x

x x


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V.A.: x 2 = 0 x = 2

( )+

=

2 12

x f x

x

O.A.: 2 1 0 1

5

4

2

2

1= + 2 y x

( )+

= =

0 1 10 ( -intercept)0 2 2 f y


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Because the numerator has no real zeros, the graph doesnot have an x -intercept.

( )+

=

2 12

x f x

x

Does the graph cross theoblique asymptote?+

= +

2 12

2 x

x x

( )( )+ = +2 1 2 2 x x x + = 2 21 4 x x no


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+=

2 12

x f x

x

( )+

=

24 14

4 2 f =

172

( ) ( ) +

=

21 11

1 2 f = =

2 23 3

( ) ( ) +=

2

6 166 2

f = 374


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Graphing Rational FunctionsAs mentioned earlier, a rational function must be

defined by an expression in lowest terms before we candetermine asymptotes or anything else about the graph.A rational function that is not in lowest terms usuallyhas a hole, or point of discontinuity , in its graph.


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Graphing Rational FunctionsExample: Graph

When you simplify a rational function, you have totake into account any values of x for which thefunction is not defined.

( )

=

2 42

x f x

x

( ) ( )( )( )

+=

2 2

2

x x f x

x

= + 2 ( 2) x x


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Graphing Rational FunctionsExample: Graph ( )

=

2 42

x f x

x


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Homework College Algebra

Page 372: 37-48, 65-90 ( 5)

obj. 9 rational functions (presentation)

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