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
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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 an1 x ... a1 x a0
R( x )
m
m1
bm x bm1 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