RATIONALFUNCTIONS

A rational function is a function of the form:

xqxpxR where p and q

are polynomials

xqxpxR

What would the domain of a rational function be?

We’d need to make sure the denominator 0

xxxR

35 2

Find the domain. 3: xx

223

xxxxH 2,2: xxx

45

12

xxxxF

If you can’t see it in your head, set the denominator = 0 and factor to find “illegal” values.

014 xx 1,4: xxx

The graph of looks like this: 2

1x

xf

Since x 0, the graph approaches 0 but never crosses or touches 0. A vertical line drawn at x = 0 is called a vertical asymptote. It is a sketching aid to figure out the graph of a rational function. There will be a vertical asymptote at x values that make the denominator = 0

If you choose x values close to 0, the graph gets close to the asymptote, but never touches it.

Let’s consider the graph x

xf 1

We recognize this function as the reciprocal function from our “library” of functions.

Can you see the vertical asymptote? Let’s see why the graph looks like it does near 0 by putting in some numbers close to 0.

10

1011

101

f

100

10011

1001

f 10

101

1101

f 100

1001

1100

1

f

The closer to 0 you get for x (from positive direction), the larger the function value will be Try some negatives

Does the function have an x intercept? x

xf 1

There is NOT a value that you can plug in for x that would make the function = 0. The graph approaches but never crosses the horizontal line y = 0. This is called a horizontal asymptote.

A graph will NEVER cross a vertical asymptote because the x value is “illegal” (would make the denominator 0)

x10

A graph may cross a horizontal asymptote near the middle of the graph but will approach it when you move to the far right or left

Graph x

xQ 13

This is just the reciprocal function transformed. We can trade the terms places to make it easier to see this.

31x

vertical translation,

moved up 3

x

xf 1

x

xQ 13

The vertical asymptote remains the same because in either function, x ≠ 0

The horizontal asymptote will move up 3 like the graph does.

Finding AsymptotesVERTICAL ASYMPTOTES

There will be a vertical asymptote at any “illegal” x value, so anywhere that would make the denominator = 0

4352

2

2

xxxxxR

Let’s set the bottom = 0 and factor and solve to find where the vertical asymptote(s) should be.

014 xx

So there are vertical asymptotes at x = 4 and x = -1.

If the degree of the numerator is less than the degree of the denominator, (remember degree is the highest power on any x term) the x axis is a horizontal asymptote.

If the degree of the numerator is less than the degree of the denominator, the x axis is a horizontal asymptote. This is along the line y = 0.

We compare the degrees of the polynomial in the numerator and the polynomial in the denominator to tell us about horizontal asymptotes.

43

522

xxxxR

degree of bottom = 2

HORIZONTAL ASYMPTOTES

degree of top = 1

1

1 < 2

If the degree of the numerator is equal to the degree of the denominator, then there is a horizontal asymptote at:

y = leading coefficient of top

leading coefficient of bottom

degree of bottom = 2

HORIZONTAL ASYMPTOTES

degree of top = 2

The leading coefficient is the number in front of the highest powered x term.

horizontal asymptote at:

1

2

43542

2

2

xxxxxR

12

y

43

5322

23

xxxxxxR

If the degree of the numerator is greater than the degree of the denominator, then there is not a horizontal asymptote, but an oblique one. The equation is found by doing long division and the quotient is the equation of the oblique asymptote ignoring the remainder.

degree of bottom = 2

OBLIQUE ASYMPTOTES

degree of top = 3

532 23 xxx432 xx

remainder a 5x

Oblique asymptote at y = x + 5

SUMMARY OF HOW TO FIND ASYMPTOTESVertical Asymptotes are the values that are NOT in the domain. To find them, set the denominator = 0 and solve.

To determine horizontal or oblique asymptotes, compare the degrees of the numerator and denominator.

1. If the degree of the top < the bottom, horizontal asymptote along the x axis (y = 0)

2. If the degree of the top = bottom, horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom

3. If the degree of the top > the bottom, oblique asymptote found by long division.

Problem 1

• In an inter-barangay basketball league, the team from Barangay Culiat has won 12 out of 25 games, a winning percentage of 48%. How many games should they win in a row to improve their win percentage to 60%?

From Problem No. 1

• We have seen that they need to win 8 games consecutively to raise their percentage to atleast 60%. What will be their winning percentage if they win (a) 10 games in a row (b) 15? 20? 30? 50? 100 games?

Why Should You Learn This?• Rational functions are used to model and solve

many problems in the business world. • Some examples of real-world scenarios are:

– Average speed over a distance (traffic engineers)

– Concentration of a mixture (chemist)– Average sales over time (sales manager)– Average costs over time (CFO’s)

Domain

Find the domain of 2x1f(x)

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

2 02

xx

Domain , 2 2,

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

You Do: Domain

Find the domain of 2)1)(x(x1-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

• 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 xx

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 xx 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 xx x

Examples

f xx

( )

412 f x

xx

( )

23 12

What similarities do you see between 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 xxx

( )

2 11 82x

15xg(x) 2

2

What similarities do you see between 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

13x54x5x3xf(x)

23

2x9xg(x)

2

What similarities do you see between 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 q(x).

 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

ayb

You DoFind all vertical and horizontal asymptotes of the following function

2 11

xf xx

Vertical Asymptote: x = -1

Horizontal Asymptote: y = 2

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

2

41

f xx

Vertical Asymptote: none

Horizontal Asymptote: y = 0

Oblique/Slant AsymptotesThe 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

1x xf xx

Vertical

x = 1

Horizontal

none

Slant

2

2

1 2

-2

xx x x

x x

y = x

Example

• Find all asymptotes: 2 2( )

1xf xx

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

xx

y = x + 1