Copyright © 2011 Pearson Education, Inc. Slide 4.2-1 4.2 More on Rational Functions and Graphs Asymptotes for Rational Functions Let define polynomials

Copyright © 2011 Pearson Education, Inc. Slide 4.2-1

4.2 More on Rational Functions and Graphs

Asymptotes for Rational Functions

Let define polynomials. For the rational function

defined by written in lowest terms, and for real

numbers a and b, 1. If then the line is a vertical

asymptote.

2. If then the line is a horizontal asymptote.

and p x q x

,p x

f xq x

as ,f x x a x a

as ,f x b x y b


4.2 Finding Asymptotes: Example 1

Example 1 Find the asymptotes of the graph of

Solution Vertical asymptotes: set denominator equal to 0 and solve.

.352

1)(

2

xxx

xf

3or 21

0)3)(12(

0352 2

xxxx

xx

.3 and

are asymptotes vertical theof equations The

21 xx



Horizontal asymptote: divide each term by the variable factor of greatest degree, in this case x2.

Therefore, the line y = 0 is the horizontal asymptote.

020

00200

352

11

352

1

)(

2

2

222

2

22

xx

xx

xxx

xx

xxx

xf0.approach all

3,

5,

1,

1

larger, andlarger gets As

22 xxxx

x




SolutionVertical asymptote: solve the equation x – 3 = 0.

Horizontal asymptote: divide each term by x.

.312

)(

xx

xf

asymptote. vertical theis 3 line The x

.2 line theis asymptote horizontal theTherefore, . as

212

0102

31

12

3

12

)(

yxx

x

xxx

xxx

xf



Solution

Vertical asymptote:

Horizontal asymptote:


.21

)(2

xx

xf

202 xx

r.denominato in the degree an thegreater th isnumerator theof degree

when theoccurs This .asymptote horizontal no is thereTherefore,

.undefined is 01

since asnumber realany approach not does x

21

11

2

1

)(

2

2

22

22

2

xx

x

xxx

xxx

xf



Rewrite f using synthetic division as follows:

For very large values of is close to 0, and

the graph approaches the line y = x +2. This line is an

oblique asymptote (neither vertical nor horizontal)

for the graph of the function.

25

221

)(2

xx

xx

xf

25

,x

x


4.2 Determining Asymptotes

To find asymptotes of a rational function defined by a rational

expression in lowest terms, use the following procedures:

1. Vertical AsymptotesSet the denominator equal to 0 and solve for x. If a is a zero of the denominator but not the numerator, then the line x = a is a vertical asymptote.

• Other Asymptotes Consider three possibilities:– If the numerator has lesser degree than the denominator, there is

a horizontal asymptote, y = 0 ( the x-axis).

– If the numerator and denominator have the same degree and f is

of the form

. line theis asymptote horizontal the

,0 where,)(0

0

n

n

nnn

nn

ba

y

bbxbaxa

xf


4.2 Determining Asymptotes

2. Other Asymptotes (continued)

(c) If the numerator is of degree exactly one greater than the denominator, there may be an oblique (or slant) asymptote. To find it, divide the numerator by the denominator and disregard any remainder. Set the rest of the quotient equal to y to get the equation of the asymptote.

Notes: • The graph of a rational function may have more than one vertical

asymptote, but can not intersect them.• The graph of a rational function may have only one other non-

vertical asymptote, and may intersect it.


4.2 Comprehensive Graph Criteria for a Rational Function

A comprehensive graph of a rational function will exhibits these features:

1. all intercepts, both x and y;2. location of all asymptotes: vertical, horizontal,

and/or oblique;3. the point at which the graph intersects its non-

vertical asymptote (if there is such a point);4. enough of the graph to exhibit the correct end

behavior (i.e. behavior as the graph approaches its nonvertical asymptote).


4.2 Graphing a Rational Function

Let define a rational expression in lowest terms.

To sketch its graph, follow these steps:

1. Find all asymptotes. 2. Find the x- and y-intercepts.3. Determine whether the graph will intersect its non-

vertical asymptote by solving f (x) = k where y = k is the horizontal asymptote, or f (x) = mx + b where y = mx + b is the equation of the oblique asymptote.

1. Plot a few selected points, as necessary. Choose an x-value between the vertical asymptotes and x-intercepts.

2. Complete the sketch.

)()(

)(xqxp

xf



Example Graph

Solution Step 1

Step 2 x-intercept: solve f (x) = 0

.352

1)(

2

xxx

xf

axis.- theis asymptote horizontal theand ,3 and equations have

asymptotes vertical the1, Example From

21

xxx

.1 isintercept - The 1

01

0352

12

xx

xxx

x



y-intercept: evaluate f (0)

Step 3 To determine if the graph intersects the horizontal asymptote, solve

Since the horizontal asymptote is the x-axis,

the graph intersects it at the point (–1,0).

31

isintercept - The 31

3)0(5)0(210

)0( 2

yf

.0)( xf asymptote horizontal of value-y



Step 4 Plot a point in each of the intervals determined by the x-intercepts and vertical asymptotes,

to get an

idea of how the graph behaves in each region.

Step 5 Complete the sketch. The graph approaches its asymptotes as the points become farther away from the origin.

),( and ),,1(),1,3(),3,( 21

21


4.2 Graphing a Rational Function That Does Not Intersect Its Horizontal Asymptote

Example Graph

Solution Vertical Asymptote:

Horizontal Asymptote:

x-intercept:

y-intercept:

Does the graph intersect the horizontal asymptote?

.312

)(

xx

xf

303 xx

212 yy

21or 01203

12 xxx

x

31

301)0(2

)0( f

solution. No12623

122

xx

x

x


4.2 Graphing a Rational Function That Does Not Intersect Its Horizontal Asymptote

To complete the graph of choose

points (–4,1) and .

,312

)(

xx

xf 3

13,6


4.2 Graphing a Rational Function with an Oblique Asymptote

Example Graph

Solution Vertical asymptote:

Oblique asymptote:

x-intercept: None since x2 + 1 has no real solutions.

y-intercept:

.21

)(2

xx

xf

202 xx

.2 line theis asymptote oblique

thee therefor,2

52

2

12

xyx

xx

x

.21 isintercept - theso ,2

1)0( yf


4.2 Graphing a Rational Function with an Oblique Asymptote

Does the graph intersect the oblique asymptote?

To complete the graph, choose the points

.32,1 and 2

17,4

.solution No41

221

22

2

xx

xxx


4.2 Graphing a Rational Function with a Hole

Example Graph

Solution Notice the domain of the function cannot include 2.

Rewrite f in lowest terms by factoring the numerator.

.24

)(2

xx

xf

)2(22

)2)(2(24

)(2

xxx

xxxx

xf

The graph of f is the graph of the line y = x + 2 with the exception of the point with x-value 2.

Documents

Copyright © 2011 Pearson Education, Inc. Slide 4.2-1 4.2 More on Rational Functions and Graphs Asymptotes for Rational Functions Let define polynomials