This document provides an introduction to rational functions and their graphs. It discusses how rational functions are used to model real-world scenarios in business and science. The key aspects covered include: - The definition of a rational function as a quotient of two polynomial functions. - The parent rational function 1/x and how its graph is not continuous and has asymptotes. - How transformations of the parent function, such as vertical and horizontal shifts, affect the graph. - How to find the domain of a rational function by determining where the denominator is undefined. - That vertical asymptotes occur at values in the domain where the function is undefined.

1. Rational Functions and Their
Graphs

2. 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)

3. Introduction to Rational
Functions
 What is a rational number?
 So what is an irrational number?
 A rational function has the form
( )
( )
( )
p x
f x
q x
where p and q are polynomial functions

A number that can be expressed as a fraction:
A number that cannot be expressed as a fraction: , 2

5
, 3, 4.5
2

4. Parent Function
 The parent function is
 The graph of the parent
rational function looks
like…………………….
 The graph is not continuous
and has asymptotes
1
x

5. Transformations
 The parent function
 How does this move?
1
x
1
3
x


6. Transformations
 The parent function
 How does this move?
1
x
1
( 3)
x 

7. Transformations
 The parent function
 And what about
this?
1
x
1
4
( 2)
x



8. Transformations
 The parent function
 How does this move?
1
x
2
1
x

9. Transformations
2
1
x 2
1
2
x

2
1
4
( 3)
x


2
1
( 3)
x 

10. Domain
Find the domain of 2
x
1
f(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????
-2 is excluded

11. You Do: Domain
Find the domain of
2)
1)(x
(x
1
-
x
f(x)



Denominator can’t equal 0
  
1 2 0
1, 2
x x
x
  
  
     
Domain , 2 2, 1 1,
       

12. You Do: Domain
Find the domain of 

2
x
f(x)
x 1
Denominator can’t equal 0
2
2
1 0
1
x
x
 
 
 
Domain ,
 

13. Vertical Asymptotes
At 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.
2
x
1
f(x)

 2
x  
2)
1)(x
(x
1
-
x
f(x)


 1, 2
x x
   


2
x
f(x)
x 1
none

14. Vertical Asymptotes
The figure below shows the graph of 2
x
1
f(x)


The equation of the vertical asymptote is 2
x  