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

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

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

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

Nasty Bottoms Vertical Asymptotes What do we do if the factoring is not possible? Ex 4 6 3 𝑥 2 −7𝑥−11 find the vertical Asymptotes

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.

Find the horizontal asymptote EX 5 12𝑥 2 −5 4𝑥 2 −7𝑥+10 EX 6 𝑥 4 −5 𝑥 6 −7𝑥+10 EX 7 𝑥 5 −7𝑥+11 𝑥−4

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

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?

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

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.

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.