Warm up: Get 2 color pencils and a ruler

Give your best definition and one example of the following:

• Domain• Range• Ratio• Leading coefficient

Chapter 9:“Let’s Be Rational About This”

In this Unit you will learn to…

9.a: determine key characteristics of simple rational functions [9.1]

9.b: multiply and divide rational expressions [9.3]

9.c: add and subtract rational expressions [9.4]

9.a: determine key characteristics of simple rational functions [9.1]

After this lesson you will be able to…

Identify and graph a rational function.

Determine the domain and range of a rational function.

Determine the horizontal and vertical asymptotes of a rational function.

Determine how to find holes in rational functions.

What is a Rational Function?

A rational function is the ratio of two polynomial expressions.

In order for a function to be rational it must meet the following requirements:

1. f(x)=

2. p(x) and q(x) are polynomial functions.

3. q(x) 0 (The denominator cannot equal 0)

( )

( )

p x

q x

Is this function rational?

2

2

31) ( ) 2)

1 2 3

5 33) 4) ( )

4 5 7

xf x y

x x

x xy f x

x x

Quick Definition:A Rational Function MUST have an ____________

in the __________________________________________.

Graph it( )

1

xf x

x

Vertical Asymptote

The red line is called a vertical asymptote.

Vertical Asymptote: The vertical line the graph will not cross because that x value causes a zero in the denominator.

1. Are there any values of x that make the function undefined (denom = 0)?

2. To find algebraically:a) Set the denominator = 0b) Solve for x.

3. The line x= -1 is a vertical asymptote.

Identify the Vertical Asymptote

( )1

xf x

x

Identify the Horizontal Asymptote

The blue line is a horizontal asymptote.

The horizontal line the graph will not cross because some x value causes a zero in the denominator.

Identify the Domain

Domain: All the x-values that DO work in the function.

What can x equal? Are there any values that x cannot equal?

For our function:

D: __________________________

Identify the Range

Range: All the y-values that DO work for the function. What can y equal? Are there any values that y cannot

equal?

For our function:

R: ____________________________

Try this one:

1( )

2f x

x

Graph the function Find:

Vertical Asymptote:_______________ Horizontal Asymptote:_____________ Domain:________________________ Range:_________________________

Try this one:

2( )

( 1)( 3)g x

x x

Graph the function Find:

Vertical Asymptote:_______________ Horizontal Asymptote:_____________ Domain:________________________ Range:_________________________

What did we learn?