Chapter 2 6 rational functions and asymptotes
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Chapter 2.6 Rational Functions and Asymptotes. Feb. 2011. Section Objectives: Students will know how to determine the domains and find the asymptotes of rational functions. Concept. Textbook pp.142-143.

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Chapter 2.6 Rational Functions and Asymptotes

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Chapter 2 6 rational functions and asymptotes

Chapter 2.6 Rational Functions and Asymptotes

Feb. 2011


Chapter 2 6 rational functions and asymptotes

Section Objectives:Students will know how to determine the domains and findthe asymptotes of rational functions.


Concept

Concept

Textbook

pp.142-143

A rational function is a function of the form f(x) = N(x)/D(x), where N and D are both polynomials.

The domain of f is all x such that D(x) ≠ 0.

Example1 Find the domain of

The domain is


Concept1

Concept

Textbook

pp.143

II. Horizontal and Vertical Asymptotes

Let’s sketch the graph of  

Now, let’s just plug in some values of x and see what we get.


Concept2

Concept

Textbook

pp.143

II. Horizontal and Vertical Asymptotes

Let’s sketch the graph of  

Now, let’s just plug in some values of x and see what we get.


Concept3

Concept

Textbook

pp.143

II. Horizontal and Vertical Asymptotes

Let’s sketch the graph of  

Now, let’s just plug in some values of x and see what we get.

As x approaches zero from the left, y decreases without bound.

+

As x approaches zero from the right, y increases without bound.

The line x = 0 is a vertical asymptote of .


Concept4

Concept

Textbook

pp.143-146

II. Horizontal and Vertical Asymptotes

Let’s sketch the graph of  

Now, let’s just plug in some values of x and see what we get.

You can see that the graph of f

also has a horizontal asymptote, the line y = 0 . This means that the values of f (x) approach zero as x increases or decreases without bound.


Concept5

Concept

Textbook

pp.143

II. Horizontal and Vertical Asymptotes


Concept6

Concept

Textbook

pp.144

II. Horizontal and Vertical Asymptotes

Determining asymptotes is actually a fairly simple process.  First, let’s start with the rational function

where n is the largest exponent in the numerator and m is the largest exponent in the denominator.  

The graph of has vertical asymptotes at the zeros of

2. The graph of has at most one horizontal asymptote determined by comparing the degrees of .

a. If the graph of has the line (the -axis) as a horizontal asymptote.

b. If , the graph of has the line as a horizontal asymptote where is the leading coefficient of the numerator and is the leading coefficient of the denominator.

c. If , the graph of has no horizontal asymptote.


Example1

Example1

Textbook

pp.144


Example2

Example2

Textbook

pp.144

The horizontal asymptote: y= 1/2,

the vertical asymptote: x = 3/2.

No horizontal asymptote

the verticalasymptote: x = -1


Example3

Example3

Textbook

pp.144

Find all horizontal and vertical asymptotes of the graph of each rational function.

a)

b)

c)

Horizontal Asymptotes

Horizontal Asymptotes

Horizontal Asymptotes

Vertical Asymptotes

Vertical Asymptotes

Vertical Asymptotes


Practice

Practice

HW: Page 148-151; #s 1-12; 15, 19, 23, 27-45, odd.


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