Rational Functions and Rational Functions and Their GraphsTheir Graphs

Section 2.6Section 2.6

Page 326Page 326

Rational Function- a quotient of two Rational Function- a quotient of two polynomial functions in the form polynomial functions in the form

f(x) = p(x) q(x) ≠ 0

q(x) Domain:Domain:

DefinitionsDefinitions

Example 1Example 1

Find the domain of each rational functionFind the domain of each rational function

9

3)()

9)()

3

9)()

2

2

2

x

xxhc

x

xxgb

x

xxfa

Reciprocal FunctionReciprocal Function0domain ,

1)( x

xxf

)(,0 as

)(,0 as

)(, as

0)(, as

xfx

xfx

xfx

xfx

Arrow Notation

(see page 328)

0domain ,1

)(2

xx

xf

)(,0 as

)(,0 as

)(, as

)(, as

xfx

xfx

xfx

xfxArrow Notation

Use the graph to answer the Use the graph to answer the following questions.following questions.

As x → -2As x → -2--, f(x) →, f(x) → As x → -2As x → -2++, f(x) →, f(x) → As x → 2As x → 2--, f(x) →, f(x) → As x → 2As x → 2++, f(x) →, f(x) → As x → -As x → -, f(x) → , f(x) → As x → As x → , f(x) → , f(x) →

Vertical AsymptotesVertical Asymptotes

Definition: the line Definition: the line x = ax = a is a vertical asymptote of is a vertical asymptote of the graph of a function if the graph of a function if f(x)f(x) increases or decreases increases or decreases (goes to infinity) without bound as (goes to infinity) without bound as xx approaches approaches aa

Locating Vertical Asymptotes: set the denominator of Locating Vertical Asymptotes: set the denominator of your rational function equal to zero and solve for xyour rational function equal to zero and solve for x

Find the vertical asymptotes of f(x) = Find the vertical asymptotes of f(x) = x – 1 x – 1

xx22 – 4 – 4

HomeworkHomework

Page 342 #1 - 28Page 342 #1 - 28

HolesHoles

A value where the denominator of a rational function is A value where the denominator of a rational function is equal to zero does not necessarily result in a vertical equal to zero does not necessarily result in a vertical asymptote.asymptote.

If the numerator and the denominator of the rational If the numerator and the denominator of the rational function has a common factor (x – c) then the graph will function has a common factor (x – c) then the graph will have a hole at x = chave a hole at x = c

Example: f(x) = Example: f(x) = (x(x22 – 4) – 4)

x – 2 x – 2

Finding the Horizontal AsymptoteFinding the Horizontal Asymptote

n < mn < m The horizontal asymptote is y = 0The horizontal asymptote is y = 0

n = mn = m The horizontal asymptote is the ratio The horizontal asymptote is the ratio of the leading coefficientsof the leading coefficients

n > mn > m There is no horizontal asymptoteThere is no horizontal asymptote

First identify the degree (highest power) of p(x) and q(x).

f(x) = p(x) degree nq(x) degree m

and identify their leading coefficients.

253

492

2

xx

xy

xx

xy

32

12

24

86)(

23

2

xx

xxxf

Find the Vertical and Horizontal Find the Vertical and Horizontal AsymptotesAsymptotes

Review Transformation of Review Transformation of FunctionsFunctions

Describe how the graphs of the following functions Describe how the graphs of the following functions are transformed from its parent function.are transformed from its parent function.

xxf

1)( 2

1)(

xxf

1)5(

1)(4. 6

3

1)(.3

31

)(2. 1

1)(.1

2

2

xxf

xxf

xxf

xxf

HomeworkHomework

Page 342 #29 - 48Page 342 #29 - 48

Graphing Rational FunctionsGraphing Rational Functions

Seven Step Strategy – page 334Seven Step Strategy – page 334

1.1. Check for symmetryCheck for symmetry

2.2. Find the interceptsFind the intercepts

3.3. Find the asymptotes – check for holesFind the asymptotes – check for holes

4.4. Plot additional points as necessaryPlot additional points as necessary

Example 6 – Graph Example 6 – Graph 1.1. SymmetrySymmetry

2.2. InterceptsIntercepts

3.3. AsymptotesAsymptotes

4.4. Plot pointsPlot points

4

3)(

2

2

x

xxf

Example – Graph Example – Graph 1.1. SymmetrySymmetry

2.2. InterceptsIntercepts

3.3. AsymptotesAsymptotes

4.4. Plot pointsPlot points

6

2)(

2

xx

xxf

Slant AsymptotesSlant Asymptotes Slant Asymptotes occur when the degree of the numerator of a Slant Asymptotes occur when the degree of the numerator of a

rational function is exactly one greater than that of the rational function is exactly one greater than that of the denominatordenominator

Note- when the degrees are the same or the denominator has a Note- when the degrees are the same or the denominator has a greater degree the function has a greater degree the function has a horizontalhorizontal asymptote. asymptote.

f x = x3+1

x2

6

4

2

-2

-4

-5 5

Line l is a slant asymptote for a function f(x) if the graph of y = f(x) approaches l as x → ∞ or as x → -∞

l

Determine the Slant AsymptoteDetermine the Slant Asymptote

Use synthetic division to Use synthetic division to find the slant asymptote find the slant asymptote then graph the functionthen graph the function

12

10

8

6

4

2

-2

5

2

132)(

2

x

xxxf

Find the Slant AsymptoteFind the Slant Asymptote

53

74)(

2

23

x

xxxf

use long division

Partner WorkPartner WorkCheck for symmetry then find the intercepts, Check for symmetry then find the intercepts, asymptotes, and holes of each rational functionasymptotes, and holes of each rational function

9

43)(.6

352

3)(.5

1

2)(.4

1

)1()(.3

12

3)(.2

3

1)(.1

2

23

2

2

2

x

xxxxV

xx

xxT

x

xxxP

x

xxh

x

xxg

xxf

HomeworkHomework

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