WARM UP

Describe the end behavior of:

Graph the function

Determine the interval(s) on which the

function is increasing and on which it is

decreasing.

1251736)( 234 xxxxxf

1)3()( 2 xxf

LESSON 3-7 GRAPHS OF RATIONAL FUNCTIONS

Objective: 1. To graph rational functions.

2. To determine vertical, horizontal and slant

asymptotes

RATIONAL FUNCTIONS

Rational functions are the

quotient of 2 polynomial functions

where h(x) ≠ 0.

The parent function is

)(

)()(

xh

xgxf

xxf

1)(

ASYMPTOTES

Asymptotes are lines  that a

graph approaches, but does not

intersect.

We will look at vertical,

horizontal and slant asymptotes.

ASYMPTOTES

A rational function can have

more than one vertical

asymptote, but it usually has

one horizontal asymptote at

most.

VERTICAL ASYMPTOTES

If g(x) and h(x) have no common factors, then f(x) has vertical asymptote(s) when h(x) = 0. Thus the graph has vertical asymptotes at the zeros of the denominator. (where the denom. is undefined.)

Rational functions are the quotient of 2 polynomial functions

where h(x) ≠ 0.)(

)()(

xh

xgxf

VERTICAL ASYMPTOTES

x=a is a vertical asymptote for f(x) if or as from either the left or the right.

)(xf

)(xf ax

EXAMPLE

For , the vertical

asymptote is x = 0

xxf

1)(

EXAMPLE

Find the vertical asymptote of

Since

the function is undefined at 1 and

-1. Thus the vertical asymptotes are

x = 1 and x = -1.

.1

2)(

2 x

xxf

)1)(1(1)( 2 xxxxh

HORIZONTAL ASYMPTOTE

To determine or prove the horizontal

asymptote:

Find the highest degree variable in

the denominator.

Divide each term in the function by

this.

HORIZONTAL ASYMPTOTE

3

13

x

xy

xxx

xxx

y3

13

highest degree variable is x

0togoesthis

x

EXAMPLE CONT’D

01

03

3y is the horizontal asymptote

SHORT CUTS

a. If the degree of g(x) is less than the degree of h(x), then the horizontal asymptote is y = 0. (this could be y= some vertical shift also)b. If the degree of g(x) is equal to the degree

of h(x), then the horizontal asymptote is

.)( oft coefficien leading

)( oft coefficien leading

xh

xgy

c. If the degree of g(x) is greater than the degree of h(x), then there is no horizontal asymptote.

)(

)()(

xh

xgxf

HORIZONTAL ASYMPTOTES Example:Find the horizontal asymptote:

.1

3)(

2 x

xxf

Since the degree of the numerator is less than the degree of the denominator, horizontal asymptote is y = 0.

Degree of numerator = 1Degree of denominator = 2

HORIZONTAL ASYMPTOTES Example:Find the horizontal asymptote: .

12

13)(

x

xxf

Degree of numerator = 1Degree of denominator = 1

Since the degree of the numerator is equal to the degree of the denominator, horizontal asymptote is . 2

3y

Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

HORIZONTAL ASYMPTOTES

Example:Find the horizontal asymptote: .

12

13)(

2

x

xxf

Degree of numerator = 2Degree of denominator = 1

EXCEPTIONS

If there are 2 vertical asymptotes, the

horizontal asymptote may or may not hold.

vert. asymp at x=.562 and -3.562

horiz. aymp at y = 1

23)(

2

2

xx

xxf

EXCEPTIONS

VERTICAL & HORIZONTAL ASYMPTOTES Practice:Find the vertical and horizontal asymptotes:

1

12)(

x

xxf

SLANT ASYMPTOTES

Slant asymptotes occur when the degree

of the numerator of a rational function is 1

more than the degree of the denominator.

Find by using polynomial long division.

ex: 2

3 1)(

x

xxf

SLANT ASYMPTOTES

Using polynomial long division will

yield

As , therefore

the slant asymptote is y = x.

xx

1

x 01

x

PRACTICE

Find the slant asymptote for

2

132)(

x

xxxf

4

3764)(

2

x

xxxf

EXCEPTIONS

Some rational functions will only have

point discontinuity instead of an

asymptote.

This occurs whenever the numerator and

the denominator share a common factor.

EXCEPTIONS

Find the asymptotes for 2

2)(

2

x

xxxf

No vertical asymptoteNo horizontal asymptoteBecause numer. & denom. have (x-2) in common there is also no slant asymptote – but a hole(point ofdiscontinuity) at x=2.

SOURCESteachers.henrico.k12.va.us/math/hcpsalgebra2/.../AII7_E_asymptotes.ppt, Sept 17,2013

WolframAlpha. Wolfram Alpha LLC, 2013. Web. 20 Sept. 2013. <http://www.wolframalpha.com>.

Find the horizontal asymptote:

x

. f xx

2 11

2

x. f x

x

3

2

12

x

. f xx x2

23

20

Exponents are the same; divide the coefficients

Bigger on Top; None

Bigger on Bottom; y=0