2.6Rational Functions

JMerrill,2010

Domain

Find the domain of 2x1f(x)

Denominator can’t equal 0 (it is undefined there)

2 0

2

x

x

Domain , 2 2,

Think: what numbers can I put in for x????

You Do: Domain

Find the domain of 2)1)(x(x

1-xf(x)

Denominator can’t equal 0

1 2 0

1, 2

x x

x

Domain , 2 2, 1 1,

You Do: Domain

Find the domain of 2

xf(x)x 1

Denominator can’t equal 02

2

1 0

1

x

x

Domain ,

Vertical AsymptotesAt the value(s) for which the domain is undefined, there will be one or more vertical asymptotes. List the vertical asymptotes for the problems below.

2x1f(x)

2x

2)1)(x(x1-xf(x)

1, 2x x

2

xf(x)x 1

none

Vertical Asymptotes

The figure below shows the graph of 2x1f(x)

The equation of the vertical asymptote is 2x

Vertical Asymptotes

Definition: The line x = a is a vertical asymptote of the graph of f(x) if

or

f x f x

as x a from either the left or the right.

Look at the table of values for 2x

1f(x)

Vertical Asymptotesx f(x)

-3 -1

-2.5 -2

-2.1 -10

-2.01 -100

-2.001 -1000

As x approaches____ from the _______,

f(x) approaches _______.

-2

left

x f(x)

-1 1

-1.5 2

-1.9 10

-1.99 100

-1.999 1000

As x approaches____ from the _______,

f(x) approaches _______.

-2right

Therefore, by definition, there is a vertical asymptote at

2x

Vertical Asymptotes Describe what is happening to x and determine if a vertical asymptote exists, given the following information:

x f(x)-4 -1.333

-3.5 -2.545

-3.1 -12.16

-3.01 -120.2

-3.001 -1200

x f(x)-2 1-2.5 2.2222

-2.9 11.837

-2.99 119.84

-2.999 1199.8

As x approaches____ from the _______, f(x) approaches _______.

As x approaches____ from the _______, f(x) approaches _______.

-3 -3

left right

Therefore, a vertical asymptote Therefore, a vertical asymptote occurs at x = -3.occurs at x = -3.

Vertical Asymptotes

Set denominator = 0; solve for x Substitute x-values into numerator. The

values for which the numerator ≠ 0 are the vertical asymptotes

Example

What is the domain? x ≠ 2 so

What is the vertical asymptote? x = 2 (Set denominator = 0, plug back into

numerator, if it ≠ 0, then it’s a vertical asymptote)

( , 2) (2, )

22 3 1( )

2

x xf x

x

You Do

Domain: x2 + x – 2 = 0 (x + 2)(x - 1) = 0, so x ≠ -2, 1

Vertical Asymptote: x2 + x – 2 = 0 (x + 2)(x - 1) = 0 Neither makes the numerator = 0, so x = -2, x = 1

( , 2) ( 2,1) (1, )

2

2

2 7 4( )

2

x xf x

x x

The graph of a rational function NEVER crosses a vertical asymptote. Why?

Look at the last example:

Since the domain is , and the vertical asymptotes are x = 2, -1, that means that if the function crosses the vertical asymptote, then for some y-value, x would have to equal 2 or -1, which would make the denominator = 0!

( , 1) ( 1,2) (2, )

2

2

2 7 4( )

2

x xf x

x x

Horizontal Asymptotes

Definition:The line y = b is a horizontal asymptote if f x b as x or x

Look at the table of values for f x 1

x 2

Horizontal Asymptotesx f(x)

1 .3333

10 .08333

100 .0098

1000 .0009

y→_____ as x→________

0

x f(x)

-1 1

-10 -0.125

-100 -0.0102

-1000 -0.001

y→____ as x→________

0

Therefore, by definition, there is a horizontal Therefore, by definition, there is a horizontal asymptote asymptote at y = 0.at y = 0.

Examples

f xx

( )

4

12f x

x

x( )

2

3 12

What relationships exists between the numerator and the denominator in each of these problems?

The degree of the denominator is larger than the degree of the numerator.

Horizontal Asymptote at y = 0

Horizontal Asymptote at y = 0

Examples

h xx

x( )

2 1

1 82x

15xg(x)

2

2

What relationships exists between the numerator and the denominator in each of these problems?

The degree of the numerator is the same as the degree or the denominator.

Horizontal Asymptote at y = 2

Horizontal Asymptote at 5

2y

Examples

13x

54x5x3xf(x)

23

2x

9xg(x)

2

What relationships exists between the numerator and the denominator in each of these problems?

The degree of the numerator is larger than the degree of the denominator.

No Horizontal Asymptote

No Horizontal Asymptote

Asymptotes: Summary1. The graph of f has vertical asymptotes at the _________ of the denominator.

 2. The graph of f has at most one horizontal asymptote, as follows:

 a)   If n < d, then the ____________ is a horizontal asymptote.

b)    If n = d, then the line ____________ is a horizontal asymptote (leading coef. over leading coef.)

c)   If n > d, then the graph of f has ______ horizontal asymptote.

zeros

line y = 0

no

ay

b

Asymptotes

Some things to note: Horizontal asymptotes describe the behavior at the

ends of a function. They do not tell us anything about the function’s behavior for small values of x. Therefore, if a graph has a horizontal asymptote, it may cross the horizontal asymptote many times between its ends, but the graph must level off at one or both ends.

The graph of a rational function may or may not cross a horizontal asymptote.

The graph of a rational function NEVER crosses a vertical asymptote. Why?

You DoFind all vertical and horizontal asymptotes of the following function

2 1

1

xf x

x

Vertical Asymptote: x = -1

Horizontal Asymptote: y = 2

You Do - AgainFind all vertical and horizontal asymptotes of the following function

2

4

1f x

x

Vertical Asymptote: none

Horizontal Asymptote: y = 0

The Last You-Do (for now :)Find all asymptotes of the following function

2 9

2

xf x

x

Vertical Asymptote: x = 2

Horizontal Asymptote: none

Slant Asymptotes

The graph of a rational function has a slant asymptote if the degree of the numerator is exactly one more than the degree of the denominator. Long division is used to find slant asymptotes.

The only time you have an oblique asymptote is when there is no horizontal asymptote. You cannot have both.

When doing long division, we do not care about the remainder.

ExampleFind all asymptotes.

2 2

1

x xf x

x

Vertical

x = 1

Horizontal

none

Slant

2

2

1 2

-2

x

x x x

x x

y = x

Example

Find all asymptotes: 2 2

( )1

xf x

x

Vertical asymptote at x = 1

n > d by exactly one, so no horizontal asymptote, but there is an oblique asymptote.

2

2

11 2

2

( 1)

1

-

xx x

x x

x

x

y = x + 1

Graphing Rational Functions

To graph a rational function, you find all asymptotes

You must show your work You must identify the domain and range You must identify the x- and/or y-intercepts You may have to “blow up” part of the

graph (Zoom:Box) to actually see how the graph fits next to the asymptote.