Math 111! Name:__!Key!_______________________________________________________________Grp#_______!GrpAct – #10!! ! ! ! Rational!Functions!1! ! ! ! !Sections!4.6!&!4.7!!Prerequisite!Skills!Key!Terms!Learning!Outcomes!• lowest!t er m s !• domain!restrictions!• Factoring!• Finding!the!degree!of!a!polynomial!• Find!the!leading!coefficient!of!a!polynomial!§ ration al!fun ction !!§ ration al!eq ua tion!§ vertical!asymptote!§ horizontal!asymptote!!§ “hole”!§ radical!equatio n! !-!Identify !v er tica l!a n d !h o riz o n ta l!as y m p t o te s,!a s !w e ll!as!roots!and!holes!for!rational!functions.!-!Solve!rational!and!radical!equations.!-Translate!radical!expressions!to!rational!exponents.!!1. A!local!manufacturing!company!has!cost!and!revenue!functions!C(x)!=!31.2 x!+!15000!and!R(x)!=!69.9 x,!where !x!is!the!number!of!units!produced.!!It!costs!$31 .20!to!produ ce !on e!un it,!and!th e!co m pa ny !has!fixe d!co sts!of!15000.!The!product!sells!as!$69.90!per!unit.!From!these!we!have!a!profit!function!of!!P(x)!=!R(x)!–!C(x)!=!69.9x!–!(31.2 x!+15000)!=!38.7x$–!15000.!! ! !The!company!calculates!their!average$profit!Pas!Total ProfitUnits Produced.!! ! !For!this!company,!the!average!profit!is!given!by! P(x) =P(x)x=38.7x − 15000x.!a. Determine!each!of!the!following,!and!explain!what!they!represent!in!the!context!of!this!problem.!!P(1000)!=!!$23,700! ! ! P(1000)=!!$23.70!! P(10000)! =!!$37.20! ! ! P(100000)=!!$38.55!!b. The!graph!ofP(x)is!given.!La bel!the!axes.!!!!c. P(x)!has!a!horizontal$asymptote!of!y!=!38.7.!Explain!wh a t!th is!a sy m p to te !represents!in!the !con tex t!of!this!pro b lem .!!!!!!d. This!function!has!an!x-in te rc ep t.!Give!the!coordinates!of!that!x-inte rc ep t ,!an d !e xp la in !why!it!occurs!in!the!context!of!the!problem.!!(358,!0)!This!is!the!Break$Even$Point.$$$i.e.!mak e !fewer!item s !and!you!loose!money,!or!make!more!and!you!make!a!profit.!At!a!production!rate!of!358!units!the!cost!is!equal!to!the!revenue.!Number of items produced Ave Profit2. A!big!part!of!learning!about!rational!functions!is!understanding!how!to!identify!asymptotes!and!intercepts.!The!last!function!we!saw!had!both!a!horizontal!and!a!vertical!asymptote.!In!this!problem!we!will!look!a!little!more!closely!at!vertical!asymptotes.!You!should!have!an!understanding!of!what!a!vertical!asymptote!is!and!why!one!occurs.!!Consider!the!function223363(2)(1)()24162(4)(2)xx x xfxxx x x-- - +==-- - +.!!a. For!what!values!of!x!will!the!numerator!of!()fx!be!zero?!!!!x = 2, − 1!!!b. For!what!values!of!x!will!the!denominator!of!()fx!be!zero?!!!!x = 4, − 2!!!c. What!is!the!domain!of!?f!!!−∞,−2( )∪−2,4( )∪4,∞( )!!!d. Identify!any!ver tic al!a s ymptote s!o f!.f!!!!x = 4!!and!!!!x = −2!!!e. Iden tif y!th e !x!and!y!inte rc ep t s!o f!t h e!g ra p h !of !.f!!x!–!intercept s:!!( 2,!0 )!!a nd !( !–!1,!0)!!y!–!intercep t:!!!!(0 ,!3/ 8 )!!!! !Fill!in! the! blanks!with!either!the!word!“numerator ”!or!“denominator.” !!If!a!rational!function!f(x)!has!a!vertical!asymptote!of!x!=!a,!then!the!___denominator___!of!f!will!be!zero!when!x!=!a.$!If!a!rational!function!f(x)!has!a!root!of!x!=!a,!then!the!___numerator___!of!f!will!be!zero!when!x!=!a.$!Note%that%if%both%the%numera tor%and%denominator%are%zero%for%the %same%value%of%x,%that% ma y%or%may%not%be%a%vertical%asymptote%of%the%func tion.%There%could%merely%b e%a%ho le%in%the%graph%of%the%function.%We%will%explore%this%more%a%little%bit%later.%!Explain!how!to!determine!if!a!rational!function!has!each!of!the!following!when%looking%at%its%formula.!-!A!horizontal!asymptote!of!y!=!0.!!!!!!-!A!horizontal!asymptote!other!than!y%=!0.!!!!!!-!No!horizontal!asymptote.!!!!!!!Check!out!the!box!on!pag e!293!if!you!need!some!help!!!Reminder!–!Vertical!asymptotes!are!vertical!lines.!They!are!identified!by!an!equation!of!the!form!x$=$a.!Reminder!–!Horizontal!asymptotes!are!horizontal!lines.!They!are!identified!by!an!equation!of!the!form!y$=$b.!3. Now!we!will!turn!our!attention!towards!horizontal!asymptotes.!Horizontal!asymptotes!can!describe!the!end!behavior!of!rational!function s.!No t!all!ration al!fu nctio ns !hav e!a!ho rizo ntal!as ym p tote .!De cide !if!each !of!the !follow in g!ha s!a!ho rizon tal!asymptote!or!if!it!will!blow!up/down!to !+/ -!infinity.!If!th e !fu n ct io n !ha s !a!h o riz o n ta l!as ymptote ,!giv e!it s!e q u at io n.! !!a. 21()352xfxxx+=+-!! H"–"asymptote:""y"="0$!!!!b. 2352()1ttgtt+-=+!! H"– "asymptote:""None""!!!!!c. 323531()24xxhxx-+=+! H"–"asymptote:""y"="5/2!!!!!!! !4. !Consider!the!rational!function.!! g(x) =−3x2− 6x + 92x3− 8x=−3(x + 3)(x − 1)2x(x − 2)(x + 2)!a. Identify!t h e!v e rtic a l!as ymptote s !o f!g.!Draw!them!in!on!the!graph!as!dotted!lines!(and!label)! V"–"Asymptotes:""x"="0,"""x"="2,""and"""x"="–"2"!!!!b. Identify!the!ho riz o n ta l!a sy mptotes !o f!g.!Draw!them!in!on!the!graph!(and!label)! ! ! H"–"asymptote:""y"="0"!!!!c. Identify!a ll!in te rc ep ts !o f!g.!Label!them!on!the!graph.!! x"–"interc e p ts:""("–3,"0)""&""(1,"0)"!! y!–!intercept :!!N o n e!!!d.