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Section 2.6

Rational FunctionsandTheir Graphs

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Rational Functions

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Rational Functions are quotients of polynomial

functions. This means that rational functions can

( )be expressed as f(x)= where p and q are

( )

polynomial functions and q(x) 0. The domain

of a rat

p x

q x

ional function is the set of all real numbers

except the x-values that make the denominator zero.

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Example

Find the domain of the rational function.2 16

( )4

xf x

x

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Example

Find the domain of the rational function.

2( )

36

xf x

x

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Vertical Asymptotes of

Rational Functions

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2

1The equation f(x)=

Vertical Asymptote onthe y-axis.

x

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Two Graphs with Vertical Asymptotes, one without

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Graphing Calculator2

f(x)=9

x

x Input the equation as you see at left.

The first graph is Connected Mode. In connectedmode, the graphing calculator plots many points

and connects the points with "curves."

In dot mode, the graphing calculator plots the

same points but does not connect them. To

change the mode on the calculator press the

MODE key then scroll down to the line that says

Connected Dot and choose the one that

you want to use.

Connected mode

Dot Mode

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Example

Find the vertical asymptote, if any, of thegraph of the rational function.

2( )

36

xf x

x

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Example

Find the vertical asymptote, if any, of thegraph of the rational function.

2( )

36

xf x

x

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Example

Find the vertical asymptote, if any, of thegraph of the rational function.

2

6( )

36

xf x

x

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2 4Consider the function f(x)= . Because the denominator is zero when

2

x=2, the function's domain is all real numbers except 2. However, there is

a reduced form of the equation in which 2 does not

x

x

cause the denominator

to be zero.

A graph with a hole

corresponding to thedenominators zero. Yourcalculator will not show the hole.

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Horizontal Asymptotes ofRational Functions

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Two Graphs with Horizontal Asymptotes, one without

Notice how the horizontalasymptote intersects the graph.

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Example

Find the horizontal asymptote, if any, of thegraph of the rational function.

2

3( )

1

xf x

x

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Example

Find the horizontal asymptote, if any, of thegraph of the rational function.

2

2

6( )

1

xf x

x

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Using Transformations toGraph Rational Functions

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Graphs of Common Rational Functions

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Transformations of Rational Functions

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Example

1 1Use the graph of f(x)= to graph g(x)= 43x x

x

y

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Example

2 2

1 1Use the graph of f(x)= to graph g(x)= 24x x

x

y

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Graphing Rational Functions

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Example5

Graph f(x)= using the 7 step strategy from

2the previous slide.

x

x

x

y

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Example2

2

2Graph f(x)= using the 7 step strategy.25

x

x

x

y

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Slant Asymptotes

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The graph of a rational function has a slant asymptote

if the degree of the numerator is one more than the

degree of denominator. The equation of the slant

asymptote can be found by division. It is the equation

of the dividend with the term containing the remainder

dropped.

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Example

2

6 2Find the slant asymptote of the function f(x)= .x xx

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Example

3

2 1Find the slant asymptote of the function f(x)= .2 2x

x x

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Applications

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The cost function, C, for a business is the sum

of its fixed and variable costs.

The average cost per unit for a company to produce

x units is the sum of its fixed and variable costs dividedby the number of units produced. The average cost

function is a rational function that is denoted by . ThusC

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Example

The Fort Myers Fishing Company discovered a better material for

making fishing reels. The fixed monthly cost is $10,000 for the costof rental of space, manufacturing equipment, as well as wages andbenefits for its employees. It costs $10 for materials to make eachfishing reel.

a.Write the cost function, C, of producing x fishing poles.

b. Write the average cost function, ,of producing x fishing poles.

c. Find and interpret (1,000), (100,000)

d. What is the horizontal asympt

C

C C

ote for the graph of the average cost

function . Describe what this represents for the company.C

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(a)

(b)

(c)

(d)

2

2

Find the vertical asymptote(s) for the graph of

4the rational function f(x)= .

16

x

x 4

46, 6

4, 4

y

x

x

y

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(a)

(b)

(c)

(d)

2

2

Find the horizontal asymptote(s) for the graph of

4the rational function f(x)= .

16

x

x 4

46, 6

4, 4

y

x

x

y

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(a)

(b)

(c)

(d)

3

23Find the horizontal asymptote for f(x)= .36x

x

3

6, 6

6, 6

y

y

x

none