Rational Functions
Lesson 9.4
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
• Given
n 1
an x  an 1 x  ...  a1 x  a0
R( x) 
m
m 1
bm x  bm1 x  ...  b1 x  b0
n
• 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
an x n
bm x m
Example
• Given
3x  8 x
r ( x)  2
5x  2 x  1
2
• Graph on calculator
 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
• Consider r ( x)  2
2x  6
• Which terms dominate as x gets large
5x
• What happens to
2x2
as x gets large?
• Note:
 Degree of denominator > degree numerator
 Previous example they were equal
When Numerator Has Larger Degree
• Try
2 x2  6
r ( x) 
5x
• As x gets large, r(x) also gets large
2
• But it is asymptotic to the line y  x
5
Summarize
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
a
 Diagonal asymptote y  x
b
n
an x
bm x m
Extra Information
n
• When n – m = 2
 Function is asymptotic to a parabola
2 x3  x 2  5 x  6
y
5x
• The parabola is
 Why?
2 2 1
y  x  x 1
5
5
an x
bm x m
Try It Out
• Consider
2x
G ( x) 
x4
 What long range behavior do you predict?
 What happens for large x (negative, positive)
x
G(x)
-100
-10
10
50
100
1000
 What happens for numbers close to -4?
x
G(x)
-4.2
-4.1
-4.01 -3.99
-3.9
-3.8
Application
• Cost to manufacture n units is
C(n) = 5000 + 50n
C ( n)
• Average cost per unit is A(n) 
n
• What is C(1)? C(1000)?
• What is A(1)? A(1000)?
• What is the trend for A(n) when n gets large?
Assignment
• Lesson 9.4
• Page 413
• Exercises 1 – 21 odd