Section 3.4Section 3.4Rational FunctionsRational Functions

Rational FunctionRational Function

A rational function is a function of the formA rational function is a function of the form

Where P and Q are polynomials and P(x) andWhere P and Q are polynomials and P(x) and

Q(x) have no factor in common and Q(x) is notQ(x) have no factor in common and Q(x) is not

equal to zero.equal to zero.

( )( )

( )P x

f xQ x

ExampleExampleConsider .Consider . Find the domain and Find the domain and graph graph ff..

Solution:Solution: When the denominator When the denominator xx + 4 = 0, we have + 4 = 0, we have xx = = 4, 4, so the only input that so the only input that results in a denominator results in a denominator of 0 is of 0 is 4. Thus 4. Thus the domain is {the domain is {xx||xx 4} or 4} or ((, , 4) 4) ( (4, 4, ).).

The graph of the function is The graph of the function is the graph of the graph of yy = 1 = 1/x/x translated to the left 4 units.translated to the left 4 units.

1( )

4f x

x

Rational FunctionsRational Functions

Different from other functions becauseDifferent from other functions because

they have they have asymptotesasymptotes..

AsymptotesAsymptotes- a line that the graph of a - a line that the graph of a

function gets closer and closer to as function gets closer and closer to as oneone

travels along that line in either travels along that line in either direction.direction.

Types of AsymptotesTypes of Asymptotes

Vertical AsymptoteVertical AsymptoteOccurs where the function is Occurs where the function is

undefined,undefined,denominator is equal to zero.denominator is equal to zero.

Form:Form:x = a x = a

where a is the zero of the where a is the zero of the denominatordenominator

**Graph never crosses ****Graph never crosses **

ExampleExampleDetermine the Determine the vertical asymptotes of vertical asymptotes of the function.the function.

Factor to find the Factor to find the zeros of the zeros of the denominator:denominator:

xx22 4 = ( 4 = (xx + 2)( + 2)(xx 2) 2)

Thus the vertical Thus the vertical asymptotes are the asymptotes are the lines lines xx = = 2 and 2 and xx = 2. = 2.

2

2 3( )

4

xf x

x

Types of AsymptotesTypes of Asymptotes

Horizontal AsymptoteDetermined by the degrees of the numerator and denominator.

Form:y = a

(next slide has rules)

** Graph can cross **

Horizontal AsymptotesHorizontal Asymptotes

Look at the rational functionLook at the rational function

If If degree of P(x) < degree of Q(x),degree of P(x) < degree of Q(x), horizontal asymptote horizontal asymptote y = 0y = 0..

If If degree of P(x) = degree of Q(x),degree of P(x) = degree of Q(x), horizontal asymptote horizontal asymptote y = y =

If If degree of P(x) > degree of Q(x),degree of P(x) > degree of Q(x), no horizontal asymptoteno horizontal asymptote

( )( )

( )

P xr x

Q x

( )

( )n

n

leading coeff P x a

leading coeff Q x b

ExampleExample

Find the horizontal asymptote: Find the horizontal asymptote:

The numerator and denominator have the The numerator and denominator have the same degree. same degree. The ratio of the leading coefficients is 6/9, The ratio of the leading coefficients is 6/9, so the line so the line yy = 2/3 = 2/3 is the horizontal is the horizontal asymptote.asymptote.

4 2

4

6 3 1( )

9 3 2

x xf x

x x

True StatementsTrue Statements

The graph of a rational function The graph of a rational function never crosses a vertical asymptote.never crosses a vertical asymptote.

The graph of a rational function The graph of a rational function might cross a horizontal asymptote might cross a horizontal asymptote but does not necessarily do so.but does not necessarily do so.

Oblique AsymptoteOblique Asymptote

Degree of P(x) Degree of P(x) >> degree of Q(x) degree of Q(x)

To find oblique asymptoteTo find oblique asymptote:: Divide numerator by denominatorDivide numerator by denominator Disregard remainderDisregard remainder Set quotient equal to y (this gives Set quotient equal to y (this gives

the equation of the asymptote)the equation of the asymptote)

Occurrence of Lines as AsymptotesOccurrence of Lines as Asymptotes

For a rational function For a rational function ff((xx) = ) = pp((xx)/)/qq((xx), where ), where pp((xx) and ) and qq((xx) have no common factors other than constants:) have no common factors other than constants:

Vertical asymptotesVertical asymptotes occur at any occur at any xx-values -values that make the denominator 0.that make the denominator 0.

The The xx-axis is the horizontal asymptote-axis is the horizontal asymptote when the degree of the numerator is less than when the degree of the numerator is less than the degree of the denominator.the degree of the denominator.

A horizontal asymptote other than the A horizontal asymptote other than the xx--axisaxis occurs when the numerator and the occurs when the numerator and the denominator have the same degree.denominator have the same degree.

Occurrence of Lines as Occurrence of Lines as Asymptotes Asymptotes continuedcontinued

An oblique asymptoteAn oblique asymptote occurs when the occurs when the degree of the numerator is 1 greater than the degree of the numerator is 1 greater than the degree of the denominator.degree of the denominator.

There can be only one horizontal asymptote There can be only one horizontal asymptote or one oblique asymptote and never both.or one oblique asymptote and never both.

An asymptote is An asymptote is not not part of the graph of the part of the graph of the function.function.

To Graph a Rational FunctionTo Graph a Rational Function1.1. Find Find vertical asymptotesvertical asymptotes. (Set denominator . (Set denominator

equal to zero)equal to zero)

2.2. Find Find horizontal asymptoteshorizontal asymptotes (compare degrees) (compare degrees)

3.3. Find the Find the x-interceptsx-intercepts. (Set y = 0) ** Rational . (Set y = 0) ** Rational function is zero if numerator is zero.**function is zero if numerator is zero.**

4.4. Find the Find the y-intercepty-intercept. (Set x = 0). (Set x = 0)

5.5. Find any Find any necessary additional pointsnecessary additional points to to determine behavior between and near vertical determine behavior between and near vertical asymptotes.asymptotes.

leading coeff P(x)

leading coeff Q(x)

•If deg. of P(x) < deg. of Q(x), horiz. asymptote y = 0

•If deg. of P(x) = deg. of Q(x), horiz. asymptote y =

•If deg. of P(x) > deg. of Q(x), no horiz. asymptote

ExampleExample

Graph .Graph .

1. 1. Find the zeros by solving: Find the zeros by solving:

The graph has vertical asymptotes at The graph has vertical asymptotes at xx = 3 = 3 and and xx = = 1/2. We sketch these with dashed 1/2. We sketch these with dashed lines.lines.

2.2.Because the degree of the numerator is less Because the degree of the numerator is less than the degree of the denominator, the than the degree of the denominator, the xx--axis, axis, yy = 0, is the horizontal asymptote. = 0, is the horizontal asymptote.

2

3( )

2 5 3

xf x

x x

2

2

2 5 3 0

2 5 3 (2 1)( 3)

x x

x x x x

The zeros are 1/2 and 3, thus the domain excludes

these values.

Example continuedExample continued3.3.To find the zeros of the numerator, we solve To find the zeros of the numerator, we solve xx + +

3 = 0 and get 3 = 0 and get xx = = 3. Thus, 3. Thus, 3 is the zero of the 3 is the zero of the function, and the pair (function, and the pair (3, 0) is the 3, 0) is the xx-intercept.-intercept.

4.4.We find We find ff(0):(0):

Thus (0, Thus (0, 1) is the 1) is the yy-intercept. -intercept.

2

0 3(0)

2(0) 5(0) 3

31

3

f

Example continuedExample continued

5. 5. We find other We find other function values to function values to determine the determine the general shape of the general shape of the graph and then draw graph and then draw the graph.the graph.

7/97/944

1122

2/32/311

1/21/211

ff((xx))xx

More ExamplesMore Examples

Graph the following functions.Graph the following functions.

a) a)

b) b)

3( )

2

xf x

x

2

2

8( )

9

xf x

x

3( )

2

xf x

x

Graph aGraph a

Vertical AsymptoteVertical Asymptote

xx = = 22Horizontal AsymptoteHorizontal Asymptote

yy = 1 = 1

xx-intercept-intercept

(3, 0)(3, 0)

y-y-interceptintercept

(0(0, , 3/2)3/2)

2

2

8( )

9

xf x

x

Graph bGraph b

Vertical AsymptoteVertical Asymptote

xx = = 3, 3, xx = 3 = 3

Horizontal AsymptoteHorizontal Asymptote

yy = = 11xx-intercepts-intercepts

((2.828, 0)2.828, 0)

y-y-interceptintercept

(0(0, , 8/9)8/9)