Transcript Rational Functions and Models Lesson 4.6 Definition  Consider a function which is the quotient of two polynomials P( x) R( x)  Q( x )  Example: 2500  2

Rational Functions
and Models
Lesson 4.6
Definition

Consider a function which is the quotient of two
polynomials
P( x)
R( x) 
Q( x )

Example:
2500  2 x
r ( x) 
x
Both polynomials
Long Run Behavior
n 1
an x  an1 x  ...  a1 x  a0
R( x ) 
m
m1
bm x  bm1 x  ...  b1 x  b0
n

Given

The long run (end) behavior is determined by
the quotient of the leading terms


Leading term dominates for
large values of x for polynomial
Leading terms dominate for
the quotient for extreme x
n
an x
bm x m
Example
3x  8 x
r ( x)  2
5x  2 x  1

Given

Graph on calculator

2
Set window for -100 < x < 100, -5 < y < 5
Example

Note the value for a large x
2
3x
2
5x

How does this relate to the leading terms?
Try This One
5x
r ( x)  2
2x  6

Consider

Which terms dominate as x gets large

What happens to

Note:


5x
as x gets large?
2
2x
Degree of denominator > degree numerator
Previous example they were equal
When Numerator Has Larger
Degree
2 x2  6
r ( x) 
5x

Try

As x gets large, r(x) also gets large

But it is asymptotic to the line
2
y x
5
Summarize
n
Given a rational function with
leading terms
 When m = n
a
 Horizontal asymptote at
b
 When m > n


Horizontal asymptote at 0
When n – m = 1

Diagonal asymptote
a
y x
b
an x
bm x m
Vertical Asymptotes

A vertical asymptote happens when the function
R(x) is not defined
P( x)
 R( x)
 This happens when the
Q( x )
denominator is zero

Thus we look for the roots of the denominator
x2  9
r ( x)  2
x  5x  6

Where does this happen for r(x)?
Vertical Asymptotes

Finding the roots of
the denominator
x2  5x  6  0
( x  6)( x  1)  0
x  6 or x  1

View the graph
to verify
x2  9
r ( x)  2
x  5x  6
Zeros of Rational Functions
P( x)
 R ( x )  0  P( x)  0
Q( x )

We know that

So we look for the zeros of P(x), the
numerator

Consider


x2  9
r ( x)  2
x  5x  6
What are the roots of the numerator?
Graph the function to double check
Zeros of Rational Functions

Note the zeros of the
function when
graphed

r(x) = 0 when
x=±3
Summary


The zeros of r(x) are
x2  9
r ( x)  2
where the numerator
x  5x  6
has zeros
The vertical asymptotes of r(x)
are where the denominator has zeros
Assignment



Lesson 4.6
Page 319
Exercises 1 – 41 EOO
93, 95, 99