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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. 12 5 17 3 6 ) ( 2 3 4 x x x x x f 1 ) 3 ( ) ( 2 x x f
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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.

Jan 03, 2016

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Page 1: 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.

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

Page 2: 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.

LESSON 3-7 GRAPHS OF RATIONAL FUNCTIONS

Objective: 1. To graph rational functions.

2. To determine vertical, horizontal and slant

asymptotes

Page 3: 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.

RATIONAL FUNCTIONS

Rational functions are the

quotient of 2 polynomial functions

where h(x) ≠ 0.

The parent function is

)(

)()(

xh

xgxf

xxf

1)(

Page 4: 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.

ASYMPTOTES

Asymptotes are lines  that a

graph approaches, but does not

intersect.

We will look at vertical,

horizontal and slant asymptotes.

Page 5: 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.

ASYMPTOTES

A rational function can have

more than one vertical

asymptote, but it usually has

one horizontal asymptote at

most.

Page 6: 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.

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

Page 7: 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.

VERTICAL ASYMPTOTES

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

)(xf

)(xf ax

Page 8: 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.

EXAMPLE

For , the vertical

asymptote is x = 0

xxf

1)(

Page 9: 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.

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

Page 10: 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.

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.

Page 11: 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.

HORIZONTAL ASYMPTOTE

3

13

x

xy

xxx

xxx

y3

13

highest degree variable is x

0togoesthis

x

Page 12: 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.

EXAMPLE CONT’D

01

03

3y is the horizontal asymptote

Page 13: 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.

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

Page 14: 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.

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

Page 15: 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.

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

Page 16: 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.

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

Page 17: 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.

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

Page 18: 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.

EXCEPTIONS

Page 19: 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.

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

1

12)(

x

xxf

Page 20: 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.

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

Page 21: 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.

SLANT ASYMPTOTES

Using polynomial long division will

yield

As , therefore

the slant asymptote is y = x.

xx

1

x 01

x

Page 22: 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.

PRACTICE

Find the slant asymptote for

2

132)(

x

xxxf

4

3764)(

2

x

xxxf

Page 23: 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.

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.

Page 24: 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.

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.

Page 25: 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.

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>.

Page 26: 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.

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