1. 4.4 Rational Functions
Rational functions are the quotient of two polynomials.
Analyzing rational functions with many properties.
Find Domain
Find vertical asymptotes
Find horizontal asymptotes
Find oblique asymptotes .
Find holes in graphs.
Finding x-int and y-int

2. Concept #1 What is domain?
Ex 𝑥−3 𝑥+9 𝑥 2 −25

3. Ex #2
What if the bottom is not nice? What is domain?
𝑥 𝑥 2 −5𝑥−9

4. Concept #2 Vertical Asymptotes
The bottom polynomials that do not reduce are vertical asymptotes. Remember they are vertical lines so the equation is in the form
X=some #
Ex 3 𝑥+25 𝑥−7 𝑥 2 −4 find the vertical Asymptotes

5. Nasty Bottoms Vertical Asymptotes
What do we do if the factoring is not possible?
Ex 𝑥 2 −7𝑥−11 find the vertical Asymptotes

6. Concept #3 Horizontal Asymptotes
If the top degree is less than the bottom degree the horizontal asymptote is y=0
If the top and bottom degree are the same.Find the lead coefficient of the top and the LC of the bottom . The horizontal asymptote is a line y= 𝐿𝐶 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑜𝑝 𝐿𝐶 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑡𝑡𝑜𝑚
If the top degree is larger than the bottom there is no horizontal asymptote.

7. Find the horizontal asymptote
EX 𝑥 2 −5 4𝑥 2 −7𝑥+10
EX 6 𝑥 4 −5 𝑥 6 −7𝑥+10
EX 7 𝑥 5 −7𝑥+11 𝑥−4

8. Concept #4 Oblique Asymptotes
The only time for oblique asymptotes is when the top degree is larger than the bottom degree.
Use long division , the divided out polynomial is the oblique asymptote.
EX 8 𝑥 2 −7𝑥+11 𝑥−4

9. Concept #5 Holes
Anytime a rational function has a reduction of an expression. The function will have a hole at that point x = a number . Find the exact point of the hole by replacing the number into the reduced rational function.
EX 9 𝑥−5 𝑥 2 −7𝑥+10 Why are there holes and where are they?

10. Holes
EX 𝑥 2 +12𝑥+32 𝑥+4 Why are there holes and where are they?

11. Concept #6 Find the x-int and y-int
The x-int are what make the top zero.
The y-int is at the point x=0
EX 11 𝑥 2 −16 𝑥−8
Find x and y intercepts.

12. 4.4 Pg 290 #1-5 odd just find the domain
#7-12 all vertical asymptotes and holes only
#13-22 all Horizontal and oblique asymptotes only
#23-49odd Find all you can.