10.6 Parametric Equations

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0_006.qd /8/05 9:05 AM Page 77 Secion 0.6 77 Parameric Equaions 0.6 Parameric Equaions Wha ou should learn Evaluae ses of parameric equaions for given values of he parameer.

1 0_006.qd /8/05 9:05 AM Page 77 Secion Parameric Equaions 0.6 Parameric Equaions Wha ou should learn Evaluae ses of parameric equaions for given values of he parameer. Skech curves ha are represened b ses of parameric equaions. Rewrie ses of parameric equaions as single recangular equaions b eliminaing he parameer. Find ses of parameric equaions for graphs. Wh ou should learn i Parameric equaions are useful for modeling he pah of an objec. For insance, in Eercise 59 on page 777, ou will use a se of parameric equaions o model he pah of a baseball. Plane Curves Up o his poin ou have been represening a graph b a single equaion involving he wo variables. In his secion, ou will sud siuaions in which i is useful o inroduce a hird variable o represen a curve in he plane. To see he usefulness of his procedure, consider he pah followed b an objec ha is propelled ino he air a an angle of 5. If he iniial veloci of he objec is 8 fee per second, i can be shown ha he objec follows he parabolic pah 7 Recangular equaion as shown in Figure However, his equaion does no ell he whole sor. Alhough i does ell ou where he objec has been, i doesn ell ou when he objec was a a given poin, on he pah. To deermine his ime, ou can inroduce a hird variable, called a parameer. I is possible o wrie boh as funcions of o obain he parameric equaions 6 Parameric equaion for. Parameric equaion for From his se of equaions ou can deermine ha a ime 0, he objec is a he poin 0, 0. Similarl, a ime, he objec is a he poin, 6, so on, as shown in Figure Recangular equaion: = Parameric equaions: = = 6 + = (6, 8) 8 (0, 0) =0 = (7, 0) Curvilinear Moion: Two Variables for Posiion, One Variable for Time FIGURE 0.50 Jed Jacobsohn/Ge Images For his paricular moion problem, are coninuous funcions of, he resuling pah is a plane curve. (Recall ha a coninuous funcion is one whose graph can be raced wihou lifing he pencil from he paper.) Definiion of Plane Curve If f g are coninuous funcions of on an inerval I, he se of ordered pairs f, g is a plane curve C. The equaions f g are parameric equaions for C, is he parameer.

2 0_006.qd /8/05 9:05 AM Page Chaper 0 Topics in Analic Geomer Poin ou o our sudens he imporance of knowing he orienaion of a curve, hus he usefulness of parameric equaions. Skeching a Plane Curve When skeching a curve represened b a pair of parameric equaions, ou sill plo poins in he -plane. Each se of coordinaes, is deermined from a value chosen for he parameer. Ploing he resuling poins in he order of increasing values of races he curve in a specific direcion. This is called he orienaion of he curve. Eample Skeching a Curve Skech he curve given b he parameric equaions,. Using values of in he inerval, he parameric equaions ield he poins, shown in he able. = = 0 = FIGURE 0.5 = = 0 = FIGURE = = = = 6 = = = = = 6 = B ploing hese poins in he order of increasing, ou obain he curve C shown in Figure 0.5. Noe ha he arrows on he curve indicae is orienaion as increases from o. So, if a paricle were moving on his curve, i would sar a 0, hen move along he curve o he poin 5,. Now r Eercises (a) (b). Noe ha he graph shown in Figure 0.5 does no define as a funcion of. This poins ou one benefi of parameric equaions he can be used o represen graphs ha are more general han graphs of funcions. I ofen happens ha wo differen ses of parameric equaions have he same graph. For eample, he se of parameric equaions, has he same graph as he se given in Eample. However, b comparing he values of in Figures , ou see ha his second graph is raced ou more rapidl (considering as ime) han he firs graph. So, in applicaions, differen parameric represenaions can be used o represen various speeds a which objecs ravel along a given pah.

3 0_006.qd /8/05 9:05 AM Page 77 Eliminaing he Parameer Secion 0.6 Parameric Equaions 77 Eample uses simple poin ploing o skech he curve. This edious process can someimes be simplified b finding a recangular equaion (in ) ha has he same graph. This process is called eliminaing he parameer. Parameric equaions Solve for in one equaion. Subsiue in oher equaion. Recangular equaion Emphasize ha convering equaions from parameric o recangular form is primaril an aid in graphing. Eploraion Mos graphing uiliies have a parameric mode. If ours does, ener he parameric equaions from Eample. Over wha values should ou le var o obain he graph shown in Figure 0.5? Parameric equaions: =, = + + FIGURE 0.5 = = 0 = 0.75 Now ou can recognize ha he equaion represens a parabola wih a horizonal ais vere, 0. When convering equaions from parameric o recangular form, ou ma need o aler he domain of he recangular equaion so ha is graph maches he graph of he parameric equaions. Such a siuaion is demonsraed in Eample. Eample Eliminaing he Parameer Skech he curve represened b he equaions b eliminaing he parameer adjusing he domain of he resuling recangular equaion. Solving for in he equaion for produces which implies ha Now, subsiuing in he equaion for, ou obain he recangular equaion.. From his recangular equaion, ou can recognize ha he curve is a parabola ha opens downward has is vere a 0,. Also, his recangular equaion is defined for all values of, bu from he parameric equaion for ou can see ha he curve is defined onl when >. This implies ha ou should resric he domain of o posiive values, as shown in Figure 0.5. Now r Eercise (c).

4 0_006.qd /8/05 9:05 AM Page Chaper 0 Topics in Analic Geomer I is no necessar for he parameer in a se of parameric equaions o represen ime. The ne eample uses an angle as he parameer. To eliminae he parameer in equaions involving rigonomeric funcions, r using he ideniies as shown in Eample. θ = π sin cos sec an θ = π FIGURE 0.5 θ = π θ = 0 = cos θ = sin θ Eample Eliminaing an Angle Parameer Skech he curve represened b cos sin, 0 b eliminaing he parameer. Begin b solving for cos sin in he equaions. cos sin Solve for cos sin. Use he ideni sin cos o form an equaion involving onl. Phagorean ideni Subsiue for cos for sin. Recangular equaion From his recangular equaion, ou can see ha he graph is an ellipse cenered a 0, 0, wih verices 0, 0, minor ais of lengh b 6, as shown in Figure 0.5. Noe ha he ellipic curve is raced ou counerclockwise as varies from 0 o. Now r Eercise. cos sin 9 6 In Eamples, i is imporan o realize ha eliminaing he parameer is primaril an aid o curve skeching. If he parameric equaions represen he pah of a moving objec, he graph alone is no sufficien o describe he objec s moion. You sill need he parameric equaions o ell ou he posiion, direcion, speed a a given ime. Finding Parameric Equaions for a Graph You have been suding echniques for skeching he graph represened b a se of parameric equaions. Now consider he reverse problem ha is, how can ou find a se of parameric equaions for a given graph or a given phsical descripion? From he discussion following Eample, ou know ha such a represenaion is no unique. Tha is, he equaions, produced he same graph as he equaions,. This is furher demonsraed in Eample.

5 0_006.qd /8/05 9:05 AM Page 775 = = = = FIGURE 0.55 = = 0 = Secion 0.6 Parameric Equaions 775 Eample Finding Parameric Equaions for a Graph Find a se of parameric equaions o represen he graph of, using he following parameers. a. b. a. Leing, ou obain he parameric equaions. b. Leing, ou obain he parameric equaions. In Figure 0.55, noe how he resuling curve is oriened b he increasing values of. For par (a), he curve would have he opposie orienaion. Now r Eercise 7. Eample 5 Parameric Equaions for a Ccloid Poin ou ha a single recangular equaion can have man differen parameric represenaions. To reinforce his, demonsrae along wih pars (a) (b) of Eample he parameric equaion represenaions of he graph of using he parameers. A graphing uili can be a helpful ool in demonsraing ha each of hese represenaions ields he same graph. In Eample 5, PD represens he arc of he circle beween poins P D. Describe he ccloid raced ou b a poin P on he circumference of a circle of radius a as he circle rolls along a sraigh line in a plane. As he parameer, le be he measure of he circle s roaion, le he poin P, begin a he origin. When P is a he origin; when P is a a maimum poin a, a; when P is back on he -ais a a, 0. From Figure 0.56, ou can see ha APC 80. So, ou have 0,, sin sin80 sinapc AC a BD a cos cos80 cosapc AP a, which implies ha AP a cos Because he circle rolls along he -ais, ou know ha OD PD BD a sin. a. Furhermore, because BA DC a, ou have OD BD a a sin BA AP a a cos. So, he parameric equaions are a sin a cos. Technolog a P = (, ) ( πa, a) Ccloid: = a( θ sin θ), = a( cos θ) (π a, a) Use a graphing uili in parameric mode o obain a graph similar o Figure 0.56 b graphing he following equaions. X T T sin T Y T cos T a O A FIGURE 0.56 B θ C D πa (πa, 0) Now r Eercise 6. πa (πa, 0)

6 0_006.qd /8/05 :08 AM Page Chaper 0 Topics in Analic Geomer 0.6 Eercises VOCABULARY CHECK: Fill in he blanks.. If f g are coninuous funcions of on an inerval I, he se of ordered pairs f, g is a C. The equaions f g are equaions for C, is he.. The of a curve is he direcion in which he curve is raced ou for increasing values of he parameer.. The process of convering a se of parameric equaions o a corresponding recangular equaion is called he. PREREQUISITE SKILLS REVIEW: Pracice review algebra skills needed for his secion a Consider he parameric equaions. (a) Creae a able of - -values using 0,,,,. (b) Plo he poins, generaed in par (a), skech a graph of he parameric equaions. (c) Find he recangular equaion b eliminaing he parameer. Skech is graph. How do he graphs differ?. Consider he parameric equaions cos sin. (a) Creae a able of - -values using, 0,,. (b) Plo he poins, generaed in par (a), skech a graph of he parameric equaions. (c) Find he recangular equaion b eliminaing he parameer. Skech is graph. How do he graphs differ? In Eercises, (a) skech he curve represened b he parameric equaions (indicae he orienaion of he curve) (b) eliminae he parameer wrie he corresponding recangular equaion whose graph represens he curve. Adjus he domain of he resuling recangular equaion if necessar cos. cos sin sin, 5. sin 6. cos cos sin 7. cos 8. cos sin sin 9. e 0. e e e.. ln ln In Eercises, deermine how he plane curves differ from each oher.. (a) (b) cos cos (c) e (d) e e e. (a) (b) (c) sin (d) e sin e In Eercises 5 8, eliminae he parameer obain he sard form of he recangular equaion. 5. Line hrough,, :, 6. Circle: 7. Ellipse: h r cos, h a cos, k r sin k b sin 8. Hperbola: h a sec, k b an In Eercises 9 6, use he resuls of Eercises 5 8 o find a se of parameric equaions for he line or conic. 9. Line: passes hrough 0, 0 6, 0. Line: passes hrough, 6,. Circle: cener:, ; radius:

7 0_006.qd /8/05 9:05 AM Page 777 Secion 0.6 Parameric Equaions 777. Circle: cener:, ; radius: 5. Ellipse: verices: ±, 0; foci: ±, 0. Ellipse: verices:, 7,, ; foci: (, 5,, 5. Hperbola: verices: ±, 0; foci: ±5, 0 6. Hperbola: verices: ±, 0; foci: ±, 0 In Eercises 7, find a se of parameric equaions for he recangular equaion using (a) (b) In Eercises 5 5, use a graphing uili o graph he curve represened b he parameric equaions. 5. Ccloid: sin, cos 6. Ccloid: cos 7. Prolae ccloid: 8. Prolae ccloid: sin, cos cos 9. Hpoccloid: cos, sin 50. Curae ccloid: 8 sin, 8 cos 5. Wich of Agnesi: co, sin 5. Folium of Descares: In Eercises 5 56, mach he parameric equaions wih he correc graph describe he domain range. [The graphs are labeled (a), (b), (c), (d).] (a) sin, sin,, (b) 5. Lissajous curve: cos, sin 5. Evolue of ellipse: 55. Involue of circle: cos, 6 sin cos 56. Serpenine curve: Projecile Moion A projecile is launched a a heigh of h fee above he ground a an angle of wih he horizonal. The iniial veloci is v 0 fee per second he pah of he projecile is modeled b he parameric equaions v 0 cos In Eercises 57 58, use a graphing uili o graph he pahs of a projecile launched from ground level a each value of v 0. For each case,use he graph o approimae he maimum heigh he range of he projecile. 60, 57. (a) v 0 88 fee per second (b) v 0 fee per second (c) v 0 88 fee per second (d) v 0 fee per second 58. (a) v 0 60 fee per second (b) v 0 00 fee per second (c) v 0 60 fee per second (d) v 0 00 fee per second 60, 5, 5, 5, 5, 0, 0, sin sin cos co, sin cos h v 0 sin Spors The cener field fence in Yankee Sadium is 7 fee high 08 fee from home plae. A baseball is hi a a poin fee above he ground. I leaves he ba a an angle of degrees wih he horizonal a a speed of 00 miles per hour (see figure). Model I f θ 7 f 08 f No drawn o scale (c) (d) (a) Wrie a se of parameric equaions ha model he pah of he baseball. (b) Use a graphing uili o graph he pah of he baseball when Is he hi a home run? (c) Use a graphing uili o graph he pah of he baseball when Is he hi a home run? (d) Find he minimum angle required for he hi o be a home run. 5..

8 0_006.qd /8/05 9:05 AM Page Chaper 0 Topics in Analic Geomer 60. Spors An archer releases an arrow from a bow a a poin 5 fee above he ground. The arrow leaves he bow a an angle of 0 wih he horizonal a an iniial speed of 0 fee per second. (a) Wrie a se of parameric equaions ha model he pah of he arrow. (b) Assuming he ground is level, find he disance he arrow ravels before i his he ground. (Ignore air resisance.) (c) Use a graphing uili o graph he pah of he arrow approimae is maimum heigh. (d) Find he oal ime he arrow is in he air. 6. Projecile Moion Eliminae he parameer from he parameric equaions v 0 cos for he moion of a projecile o show ha he recangular equaion is 6 sec v 0 an h. 6. Pah of a Projecile The pah of a projecile is given b he recangular equaion (a) Use he resul of Eercise 6 o find h, v 0,. Find he parameric equaions of he pah. (b) Use a graphing uili o graph he recangular equaion for he pah of he projecile. Confirm our answer in par (a) b skeching he curve represened b he parameric equaions. (c) Use a graphing uili o approimae he maimum heigh of he projecile is range. 6. Curae Ccloid A wheel of radius a unis rolls along a sraigh line wihou slipping. The curve raced b a poin P ha is b unis from he cener b < a is called a curae ccloid (see figure). Use he angle shown in he figure o find a se of parameric equaions for he curve. a P θ b (0, a b) a ( πa, a + b) πa h v 0 sin 6 πa 6. Epiccloid A circle of radius one uni rolls around he ouside of a circle of radius wo unis wihou slipping. The curve raced b a poin on he circumference of he smaller circle is called an epiccloid (see figure). Use he angle shown in he figure o find a se of parameric equaions for he curve. Snhesis True or False? In Eercises 65 66, deermine wheher he saemen is rue or false. Jusif our answer. 65. The wo ses of parameric equaions,, 9 have he same recangular equaion. 66. The graph of he parameric equaions is he line. 67. Wriing Wrie a shor paragraph eplaining wh parameric equaions are useful. 68. Wriing Eplain he process of skeching a plane curve given b parameric equaions. Wha is mean b he orienaion of he curve? Skills Review In Eercises 69 7, solve he ssem of equaions u 7v 9w u v w 7 a b c 8 a b c a b 9c 6 8u v w 0 In Eercises 7 76, find he reference angle, skech in sard posiion (, ) θ 5 0 6

10.1 EXERCISES. y 2 t 2. y 1 t y t 3. y e

10.1 EXERCISES. y 2 t 2. y 1 t y t 3. y e 66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES SLUTIN We use a graphing device o produce he graphs for he cases a,,.5,.,,.5,, and shown in Figure 7. Noice ha all of hese curves (ecep he case a ) have