1. Rational Functions and Models
Lesson 4.5

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. Vertical Asymptotes
A vertical asymptote happens when the function R(x) is not defined
This happens when the denominator is zero
Thus we look for the roots of the denominator
Where does this happen for r(x)?

10. Vertical Asymptotes Finding the roots of the denominator
View the graph to verify

11. Zeros of Rational Functions
We know that
So we look for the zeros of P(x), the numerator
Consider
What are the roots of the numerator?
Graph the function to double check

12. Zeros of Rational Functions
Note the zeros of the function when graphed
r(x) = 0 when x = ± 3

13. Summary The zeros of r(x) are where the numerator has zeros
The vertical asymptotes of r(x) are where the denominator has zeros

14. Assignment
Lesson 4.5A
Page 297
Exercises 1 – 61 Odd

15. This is a power function
Direct Variation
The variable y is directly proportional to x when: y = k * x
(k is some constant value)
Alternatively
As x gets larger, y must also get larger
keeps the resulting k the same
This is a power function

16. Direct Variation Example:
The harder you hit the baseball
The farther it travels
Distance hit is directly proportional to the force of the hit

17. Direct Variation Suppose the constant of proportionality is 4
Then y = 4 * x
What does the graph of this function look like?

18. Inverse Variation The variable y is inversely proportional to x when
Alternatively y = k * x -1
As x gets larger, y must get smaller to keep the resulting k the same

19. Inverse Variation
Example: If you bake cookies at a higher temperature, they take less time
Time is inversely proportional to temperature

20. Inverse Variation Consider what the graph looks like
Let the constant or proportionality k = 4
Then

21. Assignment
Lesson 4.5B
Page 299
Exercises 63 – 91 odd