Section 3.5Rational Functions

and Their Graphs

Rational Functions

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.

Example

Find the domain of the rational function.

2 16( )

4

xf x

x

Example

Find the domain of the rational function.

2( )

36

xf x

x

Vertical Asymptotes of

Rational Functions

x

y1The equation f(x)=

xVertical Asymptote on

the y-axis.

2

1The equation f(x)=

Vertical Asymptote on

the y-axis.

x

Two Graphs with Vertical Asymptotes, one without

Graphing Calculator2

f(x)=9

x

x Input the equation as you see at left.

The first graph is Connected Mode. In connected

mode, 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

Example

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

2( )

36

xf x

x

Example

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

2( )

36

xf x

x

Example

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

2

6( )

36

xf x

x

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

2x=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 the denominator’s zero. Your calculator will not show the hole.

Horizontal Asymptotes of Rational Functions

Two Graphs with Horizontal Asymptotes, one without

Notice how the horizontal asymptote intersects the graph.

Example

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

2

3( )

1

xf x

x

Example

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

2

2

6( )

1

xf x

x

Using Transformations to Graph Rational Functions

Graphs of Common Rational Functions

Transformations of Rational Functions

Example

1 1Use the graph of f(x)= to graph g(x)= 4

3x x

x

y

Example

2 2

1 1Use the graph of f(x)= to graph g(x)= 2

4x x

x

y

Graphing Rational Functions

Example5

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

the previous slide.

x

x

x

y

Example2

2

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

25

x

x

x

y

Slant Asymptotes

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.

Example

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

x x

x

Example

3

2

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

2 2

x

x x

Applications

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 divided

by the number of units produced. The average cost

function is a rational function that is denoted by . ThusC

Example

The Fort Myers Fishing Company discovered a better material for making fishing reels. The fixed monthly cost is $10,000 for the cost of rental of space, manufacturing equipment, as well as wages and benefits for it’s employees. It costs $10 for materials to make each fishing 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

(a)

(b)

(c)

(d)

2

2

Find the vertical asymptote(s) for the graph of

4the rational function f(x)= .

16

x

x 4

4

6, 6

4, 4

y

x

x

y

(a)

(b)

(c)

(d)

2

2

Find the horizontal asymptote(s) for the graph of

4the rational function f(x)= .

16

x

x 4

4

6, 6

4, 4

y

x

x

y

(a)

(b)

(c)

(d)

3

2

3Find the horizontal asymptote for f(x)= .

36

x

x

3

6, 6

6, 6

y

y

x

none