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# Class Note for MATH 1310 at UH 3

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Date Created: 02/06/15

Seclinn 5 Quadnlic Func nns Page 2 at u Example 1 For the quadratre fuhehOhr form sketeh the graph and nd the maxrrhurh or rhrhrrhurh value x2 2x 3 express the fuhehor rh standard Snllltinn tn Example 1 xex22x3 2 x22x1371 Campletethzsquarefmxz2xbyaddng 2 e1 x12 2 The fuhetror rh standard form xsfx x 12 2 The vertex rs the pdrht 712 Srhee the edemereht 0fo rs posrhve the vertex rs the lowest porht on the graph The rhrhrrhurh value offs71 2 The fuhetroh has no maxrmum value The graph ofthe functionx x122 ear be obtarhed from the graph ofthe fuhehor gx x2 by shrfhhg le 1 mt and ther shrmhg upward 2 uhrts Seclinn 35 Quadnlic mphhas Page 3 a u Example 2 For the quadratre fuhehOhr 72x24x3 express the fuhehor rh standard form sketeh the graph and nd the maxrrhurh or rhrhrrhurh value shhrtihh tn Example 2 fx 2x24x3 2x272x 3 7202 e 2x1 3 2 csrhphte the square hr x2 7 2x hyrddhg 72 e 72071 5 The fuhetror rh standard form xsfx 72x 71 5 The vertex rs the porht 15 Srhee the edemereht on rs hegatrve the vertex rs the hrghestporht on thegraph The maxrrhurh value of xsl 5 The fuhetroh has no rmmmum value The graph ofthe functionx 72071 5 ear be obtamed from the graph of the funcu on gr x2 by shrmhg rrght 1 mt streetehrhg the graph vertreany by a factor of 2 re eetrhg m the xraxxs and ther shrmhg upward 5 uhrts 15 fx72x24x3 Seclinn 5 Quadnlic Func nns Pageuru Ifa quadratle functlon ls glveh m the formx axz bx e thehthe vertex h k ear be foundby uslng the formulashri and k 1 2a 211 Example 3 Gwen the quadraue fuheuOhx x2 4M3 fmdthe vertex audthe maxlmum ormlmmum value Snlntinn tn Example 3 3 2417 24 1643 The vertex ls the pomt x slhee the eoemeleht 0fo rs hegatlve the vertex ls the hlghestpomt on the graph The maxlmum value off ls 7 3 3 2 4 Example4 whte the quadraue fuhetmhx x2 2x7 3 m standard form by eompleuhg the square ldehufy the vertex and the maxlmum ormlmmum value Theh sketeh the graph Snlntinn tn Example 4 Complete the square 0er 2x by addlng 2T x 2 2xe3 x 2xle3el xealfezl Seclinn 5 Quadnlic Func nns ragesuru The fuhetrorr rh standard form xsfx x 12 The vertex rs the pomt 7174 Srhee the eoemereht ofr rs posrhve the vertex rs the lowestporht on the graph The rhrhrrhurh value ofthe fuhehor rsel 4 There rs ho maxrrhurh value To graph the fuhetrorr begrh wrth the graph of gx 8 shown below Frrst shx the graph ofg 1 mt to the left fxx12 74 Seclinn 5 Quadnlic Func nns Pzge ul N Now shx 4 umts downward n ltx1gt2 4 The graph of me gwen funeaor is shown below 7174 Example 5 Wme the quadraae functionx 7 4x71 m standard form by eompleang are square Idenafy the vertex and me maxrmum ormlmmum value Then sketch the graph Snlutinn m Example 5 Seclinn 5 Quadnlic Func nns P2ge7uf Faetor out 71 from the rst two terms ltxgtrltx r4xgtr1 Complete the square forr2 an by addmg fx7x24x71 x274xrl x274x4714 x7223 The fuhetroh m standard form xsx 7x7 22 3 The vertex rs the pomt 23 Srhee the eoemereht ofr2 15 hegatave the vertex 15 the hrghestporht on the graph The mammum value ofthe fuheaoh rsz 3 There rs ho mrmmum value To graph the fuhetroh begrh wrth the graph of gx 8 shown below Seclinn 5 Quadnlic Func nns Fxrst 5m the graph ofg 2 umts to the ngm fx 7 7072 3 Nextre ectm the xraxxs fx 7072 3 Pig 3 am Seclinn 5 Quadnlic Func nns Now shl 3 unlts upward fx 70 22 3 The graph of me glveu fuueuou ls shown below 23 Example 5 Forx 72x2 9x 3 nd the vertex hJc by uslng the formulas i aqu ei Thenldentlfythemaxlmum ormlmmum 2a 211 value ofthe funeuon Snllltinn m Example 5 Page 9 MN Seclinn 5 Quadnlic Func nns Pig muru Forx 72x2 9x 3 11 72 andb 9 To nd h subsmuce a 72 andb 9 m the formula h 372 9x3 Seclinn 5 Quadnlic Func nns Page 11 at Thevertexrstheporht 39 4 8 Smce the coef can ofr 15 hegahve the vertex 15 the hrghestporht on the graph 105 The maxlmum value ofthe fuhenoh 15f There 15 no mlmmum value Example 7 Fmd a quadrade fuhedor sausfymg the grveh condmons The vertex 15 the porht 734 and 18 15 anotherpomt on the graph Snllltinn tn Example 7 Substrtute h 73 and k 4 m the standard form of a quadrade fuhedor rarrhzk arrrzz4 ax324 f0 8 We are grveh that the pomt 1 8 15 on the graph We must determme a Frrst subsumte x 1 m the standard form rarz 4 f1a1324 a424 16a4 ane1 8 substrtute 8 for1 f116a4 816a4 Seclinn 5 Quadnlic Func nns Pzge lzuru Solve the equatloh 816a4 fora 8lod4 874 lod474 Now substltute a m the standard formx mu 4 The quadratle fuhtloh lSX g0 32 4 Example 8 A eompahy nds that lts revenue for selllngx umts of one oflts products ls glveh by the quadratle fuhetloh Rx 200x70 5 where R ls glveh m dollars What ls the maxlmum revenue7 How many umts ofthls produetmustbe sold so that the maxlmum revenue ls aehleved7 Solution tn Example 8 ForRx 2ooxeo 5x2 11 70 5 andb 200 We Wlll hdthe vertex th To fmdh subsntutea 70 5 andb 200m the formula h Seclinn 5 Quadnlic Func nns Pig unru kReiR200 2a Rx 200x 0 5 1t R200 200200e 0 52002 400007 20000 0000 The vertex rs the porht 20020000 Srhee the eoemereht ofr 15 hegahve the vertex 15 the hrghestporht on the graph The maxrmum value ofthe fuhenoh xs R200 20000 Therefore 200 umts ofthe produetmustbe soldto aehreve amaxlmum revenue of 20000 Exercise Set 35 Maximum and Minimum Values For each of the quadratic functions given below a Complete the square to write the equation in b C d 10 11 12 13 14 15 16 the standard form fx ax h2 k State the coordinates of the vertex of the parabola Sketch the graph of the parabola State the maximum or minimum value of the function and state whether it is a maximum or a minimum fx x2 6x 7 fxx278x21 fxx272x fxx210x fx2x278xll fx3x2l8x15 fx7x278x79 fx 7x2 4x77 fx4x2740x115 fx5x2710x8 fx72x278x714 fx 74x2 24x7 27 fx x2 75x3 fxx27xil fx 273x74x2 fx77ci3c2 For each of the quadratic functions given below a b 17 18 Find the vertex h k of the parabola by using the formulas h 2 a and k f i State the maximum or minimum value of the function and state whether it is a maximum or a minimum fxx2712x50 fx7x2l4x710 19 20 21 22 23 24 fx72x216x79 fx3x2712x29 fxx23xl fx x2 77x 2 fx 72x2 9x 3 fx76x2x75 For each of the following problems find a quadratic function satisfying the given conditions 25 26 27 28 Vertex 2 75 39 passes through 7 70 Vertex 71 78 39 passes through 210 Vertex 5 7 39 passes through 3 4 Vertex 74 3 39 passes through 113 Answer the following 29 30 31 Two numbers have a sum of 10 Find the largest possible value of their product Jim is beginning to create a garden in his back yard He has 60 feet of fence to enclose the rectangular garden and he wants to maximize the area of the garden Find the dimensions Jim should use for the length and Width of the garden Then state the area of the garden A rocket is fired directly upwards with a velocity of 80 ftsec The equation for its height H as a function of time I is given by the function Ht716t2 801 a Find the time at which the rocket reaches its maximum height b Find the maximum height of the rocket A manufacturer has determined that their daily profit in dollars from selling x machines is given by the function Px 720050x701x2 Using this model What is the maximum daily profit that the manufacturer can expect Seeliun 5 Quadnlic runen39uns Page 1 or u Quadratic Functions A quadratic functinnfls a funetlon tlnat ean be put m tlne form fx ax bx s Where a d and are real numbers anda ls nonzero A quadratre functloncan be expressed ln tlne form fx axrh2 1t ealledtlne standard form by tlne metlnod ofcompletmg tlne square The gaph offls aparabola wrtn vertex h It Ifa gt 0 tlne parabola opens upward and tlne vertex ls tlne lowestpornt on tlne gaph The minimum value offlsh 7 1t vertex WC xaxrh2 1t 11 gt0 Ifa lt 0 tlne parabola opens downward and tlne vertex ls tlne luglnestpolnt on tlne gaph The maximum value offsh 1t Mr A xaxrh2 1t 1 lt0

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