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# Note for MATH 1310 at UH

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Date Created: 02/06/15

Seclinn u Hurimntzland Verdealshlnsufcraphs PageZuf 32 Example 1 Sketeh the graph ofthe fuhetrohx xl Do not plot pomts butmstead apply transform an ohs to the graph of a standard fuhetroh Snllltinn tn Example Start wrth the graph ofthe fuhetroh gx l x To gaph the fuhetroh x x m shrft the graph ofg upward l umt The gaph ofg ls dlsplayed below m a dotted format to serve as a gulde m graphmg the gweh fuhetrohf Example 2 Sketeh the graph ofthe fuhetrohx xal Do not plotpomts but mstead apply transform an ohs to the graph of a standard fuhetroh Snllltinn tn Example 2 Start wrth the graph ofthe fuhetroh gx l x To gaph the fuhetroh x x71 l shrft the graph ofg to the rrght l umt The gaph ofg ls dlsplayed below m a dotted format to serve as a gulde m graphmg the gweh fuhetrohf Seclinn u Hurimntzland Versealshlnsurcraphs Pageant 32 Example 3 Sketeh the graph of the fuhetrOhr 2 a 2 Do not plot porhts but rhstead apply transformatrorrs to the graph of a standard functlon Snllltinn tn Example 3 Begln wrth the graph of gx x2 sh own below Seclinn u Hurimntzland Verdealshinsurcraphs Page 32 To graph the fuhetrOhr x 72 shrlt the graph ofg 2 uhrts downward Example Sketeh the graph ofthe funtlonxx12 Do not plot porhts butrhstead apply transformatrorrs to the graph of a standard functlon Snllltinn tn Example 4 Seclinn u Hurimmzland Ver czl li sufanhs Pigesuf 32 Begm thh the graph ofgrx x2 sh own below Seclinn u Hurimntzland Versealshlnsurcraphs Page uf 32 Example 5 Sketeh the graph ofthe fuhetrOhr 5 3 Do not plot porhts but rhstead apply transformatrorrs to the graph of a standard functlon Snllltinn tn Example 5 Begln wrth the graph ofgx 5 shown below Seclinn u Hurimntzland Versealshlnsurcraphs Pzge7uf 32 The graph of the glven function ls shown below AnaEn Example 5 Sketch the graph of the fuhehohr l rule 2 Do not plotpolnts butlnstead apply transformations to the graph of a standard fuhehoh Snlutinn m Example 5 Begm wlth the graph ofgx shown below Seclinn u Hurimntzl and Veniczl Shifts 111 02th Fxrst 51 th graph ofg 3 umts to the le fx xl3j 2 mu Now shx 2 umts downward mh HJ Pige uf 32 Seclinn u Hurimntzland Versealshlnsurcraphs Pageant 32 The graph of the glven fuhehor ls shown below Example 7 The graph of a funcu on y fx ls shown below Sketch the graph ofthe functlony rel 2 Snlutinn m Example 7 5min u Hur39mmtzl and Veniczl Shifts 111 02th Fm shx the graph off 1 um to the nght y NHM h 1 1n um Now 5m upward 2 umts yltx71gt12 um39 39 Pig m ul 2 Sectnn u Hur39mmtzland Verticaishinsurcraphs Page u hr 2 Re ecting Stretching znd Shrinking hr anhs The graph of the given funehon is shown below yx712 Re ecting Stretching and Shrinking of Graphs Re ecting Graphs To gaph the functiony ex re eetthe gaph of y fx inthe xraxis To graph the functiony ex re eetthe graph of y fx mtheyraxls Vertical Stretching and Shrinking hr Graphs To gaph the function y ar 11 gt1 stretchthe graph of y x vertreauy by a factor ofa To graphthe functiony ar 0 1 lt1 shrrnkthe graph of y x vertre ally by a factor ofa Re ecting Graphs in the xix yx 5min u szbcling Sketching m Shrinh39ng uf anhs Re ecting Graphs in 2 Shrinking Graph 0 the yraxis yx 1 ltalt1 yx K Fig 1211f 2 Seernn u Relleelhrg5nelehhrgand lrinh39ngufanhs Page lznf 2 Example 1 Sketeh the graph of the funetronx x Donotplotporntsbutrnstead apply transformah ons to the graph of a standard funetron Snllltinn tn Example 1 Start wrth the gaph ofthe funetron gx l x To gaph the funetronx x l refleet the gaph ofg ln the xraxls The gaph ofg ls dlsplayed below ln a dotted format to serve as a gulde ln graphlng the glven funetronf Example Sketeh the graph of the funetronx 2 xl Donotplotporntsbutrnstead apply transform an ons to the graph of a standard funetron Snllltinn tn Example 2 Start wrth the gaph ofthe funetron gx x To gaph the funetronx zlx l streteh the gaph ofg by a faetor of 2 The gaph ofg ls dlsplayed below ln a dotted format to serve as a gulde ln graphlng the glven funetronf Secl39nn u unplugsumhhrgm lrinh39ngufanhs Pig Nut 2 Example 3 Sketch the graph of the functlonfx arE Do not plot polnts but lnstead apply transformations to the graph of a standard fuhrmh Snlutinn m Example Begln wlth the graph ofgx J shown below Secl39nn u Rememrg5netehhrgam irinh39ngufanhs Page 1511f 2 To graph the fuhetrOhr erE re eet the graph ofg m the xraxxs E Example 4 The graph of a funcn on y fx rs shown below Sketeh the graph of the fuhetrorry f7x E 393 yx Seernn u Rel39leellng smelling znd Shrinh39ng hr anhs Snlntlnn tn Example 4 To graph the functlony fer re eet the graph of ln the yams The graph of the glven funehon ls shown below yx 73 725 72 45 4 ens U Example 5 Page I uf 2 Sketeh the graph of the functlonfx 3x2 Do not plotpornts butrnsteaol apply transform an ons to the graph of a standard functlon Snlntlnn tn Example 5min u szbcling sumhing m Shrinh39ng uf anhs Pig 17 ul 2 Begm thh the graph of gx 2 shown below The graph of the gum function 5 shown below 12 z m x3x Secl39nn u Reneellng swelling and Shrinh39ng hr anhs Page IE hr 2 Example 5 Sketch the graph of the fuhetrohr 72 x ll Do not plot porhts but rhsteaol apply transform an ohs to the graph of a standard functlon Snllltinn tn Example 5 Begln wrth the graph of gx l shown below Frrst stretch the graph of g vertreally by a factor of 2 fx 42pm 5min u szbcling Sketching m Shrinh39ng uf anhs Pig 19 ul 2 Next re ect m thexraxxs 2x1 Now shm upward 1 um lt gt 2XH13 Secl39nn u Relleelhrg5netehhrgam lrinh39ngufanhs Pagemur 2 The graph of the gweh fuhetaoh ls shown below r zlrll Example 7 Sketeh the graph of the funtlonx apply transformatrohs to the graph of a standard functlon x73 7 2 Do not plot porhts butrhsteaol Snllltinn tn Example Begln wrth the graph of gx l shown below 5min u szbcling Sketching m Shrinh39ng uf anhs Fxrst 5m the graph of g 3 umts to the ngm fx f 31 7 2 Next re ectm the Hm fx17 x73 72 w M m V A Mm m m y M Pig 21 ul 2 Secl39nn u Ruining Smuhingand Shrinh39ng ufanhs Fig 22 ul 2 Even m on Func nns Now snxfc downward 2 umts n x73 72il L A Even and Odd Functions A functionx5 even xf frxx for all x m the domam off The gaph of an even funeuon 15 symmeme wnn respect to theyraxxs Even Functin Odd Functinn A funcuon 15 odd 1f fez x for all x m the domam off The gaph of an odd funenon 15 symmem thh respect to the ongm Secl39nn 4 Even m on Funninns Pig 2 hr 2 The gaph shown below s the gaph ofan even fuhemh x x 2 The gaph shown below s the gaph ofan had fuhemh fez x Example 1 Determan whetherthe fuhemhx xxs eveh odd ornexther Ifthe fuhemh s even or odd use symmetry to sketch its graph Snllltinn m Example 1 Secl39nn 4 Ever and on Funninns Page 2A 111 2 x x x rx ex2 ltrxgt Deerde whether or not the function rs even x x x ex x2 e x X x The fuhetrorr rs not even Deerde whether orrrot the fuhetrorr rs odd x 2x f7xx27x e xaex XX The fuhetrorr rs not odd Therefore the fuhetrorr rs rrerther even nor odd Example 2 Deterrmrre whether the fuhetaorr x s everr odd orrrerther Kthe x fuhetror rs even or odd use symmetry to sketeh rts graph Snlutinn tn Example 2 1 12 x x ex2 x Since7xfx the fuhetaorr rs even The domam of the fuhetror xsrw0u0w Begrr by rrrakrrrg a table of values by ehoosrrrg values ofx forwhxchx gt 0 Secl39nn 4 Even m on Funninns Pig 5 hr 2 i x 7 Plotthe pomts shown m the table and plot addmonalpomts lfnecessary to showthe pomon ofthe gaph forx gt 0 Use symmetxyto nd otherpomts on the gaph sthee the fuhetmh 15 even its gaph ls symmem with respect to the yraxxs Secl39nn 4 Even ml on Fun inns Page 26 ul 2 Sketeh the gaph Example 3 Gwen that the functlonf ts even and 3 4 ts apomt on the graph of fthd anothet potht on the graph Snllltinn tn Example 3 sthee we are gtveh that 34 ts apomt on the graph off 3 ts th the domam of andlt3 4 Fat an even functlonf at eaeh xth the domam off ex ts also m the domam andex x sthee 3 ts th the domam off sols e 3 ande3 3 4 Anothet potht on the graph of ts the potht 734 Example 4 Gtveh that the function ts odd and 275 ts apomt on the graph off that anothetpotht on the graph Snllltinn tn Example 4 Seet39mt 4 Even and odd Funninns Page 27 hr 2 sthee we ate gtveh that 275 ts apotttt on the graph off 2 ts th the domam of and2 5 Fat an odd fuhettOhf fat eaeh xth the domam off ex ts also th the domam andex x sthee 2 ts th the domam off sols 2 and 22 765 5 Anothet potht on the graph of ts the potht 2 5 Example 5 Detetmthe whethet the functionX 3x4 2x2 5 ts eveh odd at hetthet Snllltinn tn Example 5 sttbstttttte e x at x x 3x4 2x2 5 ex 3x4 242 5 3x4e2x15 x Met eaeh x th the domath teaex x then ts an even function Thus the gtveh function ts eveh Example 5 Detetmthe whethet the fuhettOhx 10 2xts eveh odd ethetthet Snllltinn tn Example 5 Secl39nn 4 Even m on Funninns Pig 23 ul 2 Subsmuce e x for x fx10 2x x10x57 2H 10e2x elox zx The 72x X Iffor each x m the domam off ex 40 than 15 an odd funeuon Thus the gwen funeuon 15 odd Example 7 Determan whether the functionx 4 2 1 even odd or neither x Snlutinn m Example 7 Subsmuce e x for x x 4 n1 1 x x 4H Hex 4e xzel x 1 74x3x277 x ppo mu 8A8 xaquau sx f ppmousx 4 4 4 gr 04 MKMK 2 4 fgltt74 4 uanawusxf waxy Zxgpltw x Tzugvawx z u a m suusum m w ma v m Exercise Set 34 Transforming Functions Matching The lefthand column contains equations that represent transformations of fx x2 Match the equations on the left with the description on the right of how to obtain the graph ofg from the graph off 1 10 11 12 go x7 42 gx x2 7 4 gx x2 4 go ltx 42 go 49 go HY gx 4x2 gltxx2 gx 7x2 74 gxx42 3 goo 4x73 4 go 4 A Re ect in the xaxis B Shift left 4 units then re ect in the yaxis C Re ect in the xaxis then shift downward 4 units D Shift right 4 units E Shift right 3 units then re ect in the x axis then shift upward 4 units F Shift upward 4 units G Re ect in the yaxis H Shift left 4 units then shift upward 3 units I Shift left 4 units J Shift downward 4 units K Stretch vertically by a factor of 4 L Shrink vertically by a factor of 4 Describe how the graph of g is obtained from the graph of f Do not sketch the graph 13 14 15 16 17 18 foo J foo x3 fxx foo x2 fxi X fx goo J 72 gx 72x53 gx75x72l gxltx32 7 2 goo x8 gc3 7x4 Describe how the graphs of each of the following functions can be obtained from the graph ofy fx 19 20 21 22 23 24 25 26 yfx1 yfxe7 yfx3 yfx3e8 y fx 2 5 y5fx1 yf77x2 yf7x7577 Standard Functions Sketch the graph of each of the following functions Plot points if necessary but then memorize the general shape of each graph 27 28 29 30 31 32 mo x2 x I x foo J No 9 mo foo 4 Sketch the graph of each of the following functions Do not plot points but instead apply transformations to the graph of a standard function 33 34 35 36 37 fx x2 3 x x75 foo 67x2 foo 24x71 fx 7307 42 7 2 Exercise Set 34 Transforming Functions 38 x pr 52 3 Determine whether each of the following functions is even odd both or neither 39 fx671 x2 55 fxx 75x 40 fX 7x1 56 fxx23x 41 fx 7x42 57 fxx42x2 4239 quot 5 1 58 fxx52x3 43 2 5 73 foo x I 59 fx2x3x275xl 44 f7 424 x Ix I 60 fx3x6iz X 45 m 4x 4 43 1 Answer the following 46 fx 7x3 75 1 61 The graphs of fx47x2 and gx 47le 47 quot x73 6 are shown below Describe how the graph ofg was obtained from the graph of f 48 x 2 fodder g quotquot2l x 4 49 m 41 3 x x X 50 fx3 76 51 fx37x2 62 The graphs of fxx371 and gxx37l 52 x 3 X1 5 are shown below Describe how the graph ofg was obtained from the graph of f Answer the following mail WEI 2 53 a If a function IS odd then it IS symmetric 2 with respect to the X x xaxis yaxis or origin 2 2 2 2 b If a function is even then it is symmetric 2 2 with respect to the xaxis yaxis or origin 54 a If a function is symmetric with respect to the Sketch the graphs of the following functions yaxis then the function is 39 9 Odd even both or neither 63 a x X2 79 b got I X2 79 l b If a function is symmetric with respect to the origin then the function is 1 Odd even both or neither 64 a f x b gx X l x another point on the graph that 2 7 7 is a point on the graph off Find 76 Suppose that y f x is an even function and point on the graph 73 6 is a point on the graph off Find another 75 Suppose that y f x is an odd function and that Answer the following 74 y fa 7 4 73 y72fxl 72 yfx 71 y 2fx 70 y fx 69 y fa 68 y fx15 67 y fX21 66 y NH 65 y fx2 iaiaL aaaauauaiai 44L4L4L44444444L of each of the following functions The graph of y fx is given below Sketch the graph Exercise Set 34 Transforming Functions Seclinn u Hurimntzland Veniczlshi sufanhs Page 1 ul 2 Horizontal and Vertical Shi s of Graphs Vertical Shifts To graphthe fuhetrorry fxc gt 0 shx the gaph of y x upward umts To gaph the fuhetrohy for c gt 0 shit the graph of y x downward umts Hnrilnntzl Shifts To gaphthe fuhetrohy for c c gt 0 shit the graph of y x to the nght umts To graphthe fuhetrohy fxc gt 0 shit the gaph of y x to the le umts Vertical Shifts c gt 0 393 yxr

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