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 domainof a rat

p xq 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( )

4xf xx

Example

Find the domain of the rational function.

2( )36xf x

x

Vertical Asymptotes of Rational Functions

x

y1The equation f(x)=x

Vertical 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 Calculator2f(x)=

9x

x Input the equation as you see at left.The first graph is Connected Mode. In connectedmode, the graphing calculator plots many pointsand 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( )36xf x

x

Example

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

2( )36xf x

x

Example

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

2

6( )36

xf xx

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

xx

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( )1xf x

x

Example

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

2

2

6( )1

xf xx

Using Transformations to Graph Rational Functions

Graphs of Common Rational Functions

Transformations of Rational Functions

Example1 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)= 24x x

x

y

Graphing Rational Functions

Example5Graph f(x)= using the 7 step strategy from

2the previous slide.

xx

x

y

Example2

2

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

x

x

y

Slant Asymptotes

The graph of a rational function has a slant asymptoteif 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 equationof the dividend with the term containing the remainder dropped.

Example2 6 2Find the slant asymptote of the function f(x)= .x xx

Example3

2

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

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 producex 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

ExampleThe 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 Cote 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)= . 16x

x 446, 64, 4

yxxy

(a)

(b)

(c)

(d)

2

2

Find the horizontal asymptote(s) for the graph of

4the rational function f(x)= . 16x

x 446, 64, 4

yxxy

(a)

(b)

(c)

(d)

3

2

3Find the horizontal asymptote for f(x)= .36x

x

36, 66, 6

yyxnone