1. Rational Functions
Lesson 9.4

2. Definition
Consider a function which is the quotient of two polynomials
Example:
Both polynomials

3. Long Run Behavior 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

4. Example Given Graph on calculator
Set window for -100 < x < 100, -5 < y < 5

5. Example Note the value for a large x
How does this relate to the leading terms?

6. Try This One Consider Which terms dominate as x gets large
What happens to as x gets large?
Note:
Degree of denominator > degree numerator
Previous example they were equal

7. When Numerator Has Larger Degree
Try
As x gets large, r(x) also gets large
But it is asymptotic to the line

8. Summarize Given a rational function with leading terms When m = n
Horizontal asymptote at
When m > n
Horizontal asymptote at 0
When n – m = 1
Diagonal asymptote

9. Extra Information When n – m = 2 The parabola is
Function is asymptotic to a parabola
The parabola is
Why?

10. Try It Out Consider What long range behavior do you predict?
What happens for large x (negative, positive)
What happens for numbers close to -4? x
-100
-10 10 50
100
1000
G(x) x
-4.2
-4.1
-4.01
-3.99
-3.9
-3.8
G(x)

11. Application Cost to manufacture n units is C(n) = 5000 + 50n
Average cost per unit is
What is C(1)? C(1000)?
What is A(1)? A(1000)?
What is the trend for A(n) when n gets large?

12. Assignment
Lesson 9.4
Page 413
Exercises 1 – 21 odd