21 name%%%%%%%%%%%%%%%%%%%%%%rational… 4.… · !mathematics!vision!project! ......

Name Rational Expressions and Functions 4.4 Mathematics Vision Project | MV P Licensed under the Creative Commons Attribution4NonCommercial4ShareAlike 3.0 Unported license Ready, Set, Go! Ready Topic: Distinguishing between proper and improper rational functions. Determine if each of the following is a proper or an improper rational function. (Hint: look at the degree of the polynomials.) 1. 2. 3. ! ! = ! ! + 3! ! + 7 7! ! − 2! + 1 ! ! = ! ! − 5! ! − 4 ! ! = 3! ! − 2! + 7 ! ! − 5 4. 5. ! ! = ! ! !!! ! !!! !"!!! ! ! = 5! ! − 4! + 4 7! ! − 2! + 3 6. Which of the above functions have the following end behavior: !" ! → ∞, ! ! → 0 !"# !" ! → −∞, ! ! → 0 7. Complete the statement: ALL proper rational functions have end behavior that ________________________________________________________________________. Set Topic: Features of rational functions. Find the xRintercept(s), yRintercept, roots, asymptotes, and end behavior of the following functions. 8. ! ! = (!!!)(!!!) (!!!)(!!!)(!!!) 9. ! ! = (! ! !!) (!!!)(!!!) ! Features of f(x): Features of g(x): 21

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Page 1: 21 Name%%%%%%%%%%%%%%%%%%%%%%Rational… 4.… · !Mathematics!Vision!Project! ... !ALL!proper!rational!functions!have!end!behavior!that!!! _____.! # Set# Topic:!Features!of!rational!functions.!!!

Name%%%%%%%%%%%%%%%%%%%%%%Rational%Expressions%and%Functions% 4.4#!

!Mathematics!Vision!Project!|!MVP!!!

Licensed!under!the!Creative!Commons!Attribution4NonCommercial4ShareAlike!3.0!Unported!license!

Ready,#Set,#Go!#

Ready#%

Topic:!Distinguishing!between!proper!and!improper!rational!functions.!

Determine!if!each!of!the!following!is!a!proper!or!an!improper!rational!function.!!

(Hint:!look!at!the!degree!of!the!polynomials.)!

1.! 2.! 3.!

! ! = !! + 3!! + 77!! − 2! + 1!

! ! = !! − 5!! − 4! ! ! = !3!! − 2! + 7!! − 5 !

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!4.! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!5.!

!!!!!!!!!!!!!!!!!!!!!!!!!! ! = !!!!!!!!!!"!!! ! ! ! = 5!! − 4! + 4

7!! − 2! + 3!!

#6.!Which!of!the!above!functions!have!the!following!end!behavior:!!!

!"!! → ∞, ! ! → 0!!"#!!"!! → −∞, ! ! → 0!!

7.!Complete!the!statement:!ALL!proper!rational!functions!have!end!behavior!that!!!

________________________________________________________________________.!

#Set#Topic:!Features!of!rational!functions.!!

Find!the!xRintercept(s),!yRintercept,!roots,!asymptotes,!and!end!behavior!of!the!following!

functions.!

8.!!! ! = ! (!!!)(!!!)(!!!)(!!!)(!!!)! ! ! 9.!!! ! = ! (!!!!)

(!!!)(!!!)!!

Features!of!f(x):* * * * * Features!of!g(x):!

Page 2: 21 Name%%%%%%%%%%%%%%%%%%%%%%Rational… 4.… · !Mathematics!Vision!Project! ... !ALL!proper!rational!functions!have!end!behavior!that!!! _____.! # Set# Topic:!Features!of!rational!functions.!!!

Name%%%%%%%%%%%%%%%%%%%%%%Rational%Expressions%and%Functions% 4.4#!

!Mathematics!Vision!Project!|!MVP!!!Licensed!under!the!Creative!Commons!Attribution4NonCommercial4ShareAlike!3.0!Unported!license!

Determine!if!each!rational!expression!is!proper!or!improper.!If!improper,!use!long!division!to!rewrite!the!rational!expressions!such!that!!!!(!)! ! = ! ! + ! !(!)!(!)!!!!!where!! ! !represents!the!quotient!and!!(!)!represents!the!remainder.!

10. !!!!!!!!!!!!!!!!! !

11. !!!!! !!!!!! !!! !

!!!!!

12. !!!!!!!!!!!!!!!!!!!!!!! !

13. !!!!!!!!!!!!" !

Go#Topic:%Simplifying%rational%expressions%Simplify%each%of%the%rational%expressions.%14.% 15.% 16.%

!! + 7! + 12!! + 5! + 6 %

%%%%%

2!! − 8!! − 4! + 4%

! + 4!! + 6! + 8%

17.% 18.% 19.%! + 34 ! ∙ !2(! − 6)! + 3 % ! + 2

! − 5 +! + 3! + 5%