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## Week 10 Notes

by: Thomas Frerks

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# Week 10 Notes 126B

Thomas Frerks
WVU
College Algebra 4-Day
***** *****STAFF***** (P)

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This is Week 10 of 126B Math notes. This explores the ideas of Rational Functions and Asymptotes
COURSE
College Algebra 4-Day
PROF.
***** *****STAFF***** (P)
TYPE
Class Notes
PAGES
13
WORDS
CONCEPTS
126B
KARMA
25 ?

## Popular in Mathematics (M)

This 13 page Class Notes was uploaded by Thomas Frerks on Monday October 26, 2015. The Class Notes belongs to 126B at West Virginia University taught by ***** *****STAFF***** (P) in Fall 2015. Since its upload, it has received 5 views. For similar materials see College Algebra 4-Day in Mathematics (M) at West Virginia University.

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Date Created: 10/26/15
Unit 4 Rational Functions and Exponential Functions Quiz and Exam Z CHAPTER 15 RATIONAL FUNCTIONS QUIZ ANQ TEST INFORMATION The material in this chapter is on 39I EX39I39 INFORMATION The material in this chapter corresponds to the following sections of your text book 52 and 53 Please read these sections and complete the assigned homework from the text that is given below and on the course syllabus Text Name Problem Nutribers Section 52 Properties ofRational 13 23 25 31 41 45 51 Functions 53 The Graph ofa 7 15 2397 33 35 51 6 Rational Function LAB INFORMATION Material from this chapter is used on the lab This information is also needed for future labs WW 1 Unit 4 Rational Functions and Exponential Functions X q 6 numerabr 1518 an c agree of e Otherwise the mction is improper M de a mh lej r 39 x77 0 m c i 1 0 De nition An asymgtote is 3 M Lag r 4mg 61555ng QOfg 42 497 39 rIt d A z An asymptote is not part of the graph of a function but it tells us information about the graph We will talk about ngfr39Ccl A JLZHL and 05 A7596 asymptotcs Iknfcd 0 Other examples and notes RC X ff 6 fc brc GLZXJ ug bn 56 11 KXCnd Arte A C Mnnaa 64010 quot m 201 5 a 74m 175 Unit 4 Rational Functions and Exponential Functions pm 0 Locatin Vertical Asvm totes A rational function ROE that is j M 12 x 1 1 quot39 39g 39 LFAMM n w 9 4 va in 0w f 809 will have a vertical asymptote at x r if 1339 c r t zero 02quot 9 Jeanne0r f X 2 amp rO Example Where will Rx x lquotx 2quot have vertical asymptOt S I M Af x 3 2 x v ITIX 4 Verh39a X a I x 2 Oxfymp b o Examglc Will Rx have any vertical asymptotcs x R Z X S X1H Xi X y 2 far 5 D39fxlx 5 N017 o gmplc Can the graph of a rational function intersect its venical asymptote 0 V yMc gagmy 0 Vf We 6 081904 5 Zero Ht Joy261402 1 5 VACaimed Ace 0 Example How manr vertical asymptotes can a rational function have Here 15 no ver 52 gt1 xx13 amp nz 0 WW agwtpzbtf 177 Unit 4 Rational Functions and Exponential Functions Section 153 Horizontal and Oblique Asymptotes of Rational Functions 0 Example Consider the reciprocal inction Rx 1 Make a chart of X the values of Rx for xvalues l 2 3 4 5 and 100 Next make a chart of the values of Rx for xvalues l 2 3 4 5 and 100 X x Rx x Rx X 1 7 1 1 2 11 2 quot1 3 l 3 4 4 IC 4 JA 5 U 5 vlr 100 100 100 Mm 00 O O 00 0 Example What is happening to the values of Rx as values of x get closer and closer to in nity or negative in nity lzCX gt Example For Rx as the xvalues approach infinity or negative infinity the yvalues approach 2 ewi a 5 IX 00 ZOO 90 0 De nition A rational function Rx has a horizontal asymptote at y b if as the xvalues approach positive or negative 00 the yvalues approach Horizontal asymptotes tell us about the e di b glgg39 C of a function 179 Unit 4 Rational Functions and Exponential Functions l 0 Example How many oblique asymptotes can a rational function 3 have Can a rational metion have both obllque and horizontal asymptotes Can a rational function have obliqt39ie and vertical asymptotes 1 me Or quot c of can eMztf J M 9Wb Maltvzha AQQWW 3 Ye axcl Awe Am39zgao Verna 15 0 Example Explore what happens when you divide one polynomial px by another polynomial qx when f 3 l The degree of px is less than the degree of qx paper I 3 yex are llAs r x 8 XL 2x Y X I A Zonfl l J I A 5 h39 Jr f lQ 2 The degree of px is equal to the degree of qx H gtlt 120K thxth x1i at F bucki 517quot V Qquot e lioL q E yctua of7 L may Imm ChewWI fanm 193 I81 Unit 4 Rational Functions and Exponential Functions now To weer HORIZONTAL OR OBLIQUE ASYMPTOTES or A RATIONAI FUNCTION Rx pxqx Degree of the T f YPe 0 numerator X d P Asymptolc HOW39 to F 1 versus e rec 0 mm C g horizontal or ASyumtc p the denominator bl o Ique qx RU is nun H Z r the degree 01 px 0 Here I 7quot 3 is less than the V 3 O R Iquot X degree of qx J n lt m The degree of Mid glariz t 59 q J K VX 5 39 ual to lh l 15 eq 9 0f 7 quotpl degree of qrx f 9 l y n m Vp 1 The degree ofpX 0 be 53 0 5 5 X L f is one more than a 014mm Wf 6 x the degree of qx Chg NAM f n m l VSXf 5 The degree ofpx X is more than one Q E90 L 1 more than the 39 7 x as degree ot qx k Unit 4 Rational Functions and Exponential Functions 0 Examgle Find the horizontal or oblique asymptotes if any of the graph of 2x4 5x2 4x lx2 2x l Def 63 Dowdr gt2 Aooe 56 tagfeel 0 Example Which of the following statements is true about Rx if W 10 x 2m n X 539 Venom Rx X0L x5x l 2 quot R has 2 xintercepts 2 vertical asymptotes and 2 horizontal L asymptotes 35 x R has no xintercepts 2 vertical asymptotes and 2 horizontal Lquot 085ng l asymptotes W WC 3 R has 2 xintereepts 2 vertical asymptotes and l oblique payee asymptote L4 R has 2 xintercepts 2 vertical asymptotes and 1 horizontal asymptote Ha z ESETNBHe ofthese b 0 Other examples and notes X jj I0 x NU ml xw X39W Y311 CMW l x Xi w A gwm emml quot91 2 399 l85 Unit 4 Rational Functions and Exponential Functions Section 15 I Basics of Rational Functions 0 De nition A Etional function is a ratio of two where the ma f YpQ m 39 Z is nonzero In other words a rational ftmction is a function of the form x Rx Q Ci where X and ZCX are polynomials with nonzero Examples Write some examples of rational Functions 6 00 in f ltXSfX lffx y C K 0041 x39 0 Example Look at the examples you wrote above What are the domains of these functions I XXi 0H 5 0 3 XX 0 xf 4 r M 0 Example Let Rx What is the domain of Rx39 0 gt6 I X 02 WW km 2 x 93100 x4 X t Jarxzat 0 De nition A rational tnction is groper if 0f I74 Unit 4 Rational Functions and Exponential Functions Section I 5 2 Vertical Asyngptotes of Rational Functions 0 Example Consider the reciprocal function Rx 1 Make a chart of x the values of Rx for xvalues 2 1 12 U3 U4 and 15 Next make a chart of the values of Rx for xvalues 2 1 12 1X3 l4 and 15 x x Rx x Rx X 2 101 2 72 1 1 1 1 12 01 42 32 13 3 13 a 14 L7 14 L US 5 l5 5 0 00 oo 0 0 Example What is happening to the values of Rx as values of x get closer and closer to zero 71c 55 c339 r end layff in mafmyvce 0 Example For Rx 1 as the xvalues approach 0 the y values I 2 If a tiger bccomc U jot ea C 2 gavecquot A t M r MCI 0 De nition A rational function Rx has a vertical asymptote at x r if as the xvalues approach f the yvalues become Maison64 I r quot 00 W 00 176 Unit 4 Rational Functions and Exponential Functions 0 Other examgles and nores X 1 X Wow xii204 xw H U d 0 I 3 XX JXJ 9 X 393 quot l tht r a afw bfc39r XampX j R my Uri Um X27 X I 3912 5 Afry YI armJoz e 178 Unit 4 Rational Functions and Exponential Functions 0 Example Can the graph of a rational mction intersect its horizontal asymptote 0 Example How many horizontal asymptotes can a rational function have 37 Glee5 Come None Or one E c UI Of 0 Example Write some examples of rational functions that have horizontal asymptotcs 39 1 JX rm 2 I Am 1 a 509 4 X 93 WW1 405 7 YJZO 530 gal o g nition If an asymptote is neither vertical nor horizontal it is called 0 05 Age or s f nCC Oblique asymptotes also have to do with the e 94 b66tamp439Q of a function 0 What is an example of a function with an oblique asymptote L 1 Lon A00 5 x V V X 73 Dev3km xhx w Example Can a graph intersect its oblique asymptote 180 Unit 4 Rational Functions and Exponential I unctions 3 The degree of px is one more than the degree of qx 2 an 2 X X3 agree 2 39 dwzr ran f f X 1 1 6 ay66 av l rm m 6 j z X Ski x fox fox J DA ve J7lt 3 bx 26 fx H e I i ri e 4 The degree of px is greater than the degree of qx by two or more M Aari39zmG 0r 0 de 2 00 051 x 06369 5 L1 X J Loaf AzJ a 7 ne 50 10 hanW 0r a rque asympoj 0 coating Horizontal and Obligue Asymptotgs Let Q px auxquot am a and qx bmxm l but bf be two nonzero polynomials Fill in this chart about nding the Pix horizontal or oblique asymptotcs of the rational function Rx q x 182 Unit 4 Rational Functions and Exponentian Functions 0 39xample Find the horizontal or oblique asymptotes if any of the graph of 4x 1m 2x 1 21 Del 1 39 0 AOrI Zma 0 Example Find the horizontal or oblique asymptotcs if any of the graph 58 4x 1 l 738 2x I 5 iii ef Dot 9A 77 Awnam al 0 Example Find the horizontal or oblique asymptotes if any of the graph of2x3 5x24x lx22x l 051409 c X9 l X Xjfxlcfy aiiii if 9x x 4 ampx iZOLX Q 051406 l 184

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