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333202 1006.qxd12/8/059:05 AMPage 771Section 10.6771Parametric Equations10.6 Parametric EquationsWhat you should learn Evaluate sets of parametricequations for given valuesof the parameter. Sketch curves that arerepresented by sets ofparametric equations. Rewrite sets of parametricequations as singlerectangular equations byeliminating the parameter. Find sets of parametricequations for graphs.Why you should learn itParametric equations are usefulfor modeling the path of anobject. For instance, in Exercise59 on page 777, you will use aset of parametric equations tomodel the path of a baseball.Plane CurvesUp to this point you have been representing a graph by a single equationinvolving the two variables x and y. In this section, you will study situations inwhich it is useful to introduce a third variable to represent a curve in the plane.To see the usefulness of this procedure, consider the path followed by anobject that is propelled into the air at an angle of 45 . If the initial velocity of theobject is 48 feet per second, it can be shown that the object follows theparabolic pathy x2 x72Rectangular equationas shown in Figure 10.50. However, this equation does not tell the whole story.Although it does tell you where the object has been, it doesn’t tell you when theobject was at a given point x, y on the path. To determine this time, you canintroduce a third variable t, called a parameter. It is possible to write both x andy as functions of t to obtain the parametric equationsx 24 2ty 16t 2Parametric equation for x 24 2t.Parametric equation for yFrom this set of equations you can determine that at time t 0, the object is atthe point 0, 0 . Similarly, at time t 1, the object is at the point 24 2, 24 2 16 , and so on, as shown in Figure 10.50.yRectangular equation:2y x x729Parametric equations:x 24 2ty 16t 2 24 2tt 3 24(36, 18)18(0, 0)t 0t 3 22(72, 0)x9 18 27 36 45 54 63 72 81Curvilinear Motion: Two Variables for Position, One Variable for TimeFIGURE 10.50Jed Jacobsohn/Getty ImagesFor this particular motion problem, x and y are continuous functions of t,and the resulting path is a plane curve. (Recall that a continuous function is onewhose graph can be traced without lifting the pencil from the paper.)Definition of Plane CurveIf f and g are continuous functions of t on an interval I, the set of orderedpairs f t , g t is a plane curve C. The equationsx f t andy g t are parametric equations for C, and t is the parameter.

333202 1006.qxd77212/8/059:05 AMChapter 10Page 772Topics in Analytic GeometryPoint out to your students the importance of knowing the orientation of acurve, and thus the usefulness ofparametric equations.Sketching a Plane CurveWhen sketching a curve represented by a pair of parametric equations, you stillplot points in the xy-plane. Each set of coordinates x, y is determined from avalue chosen for the parameter t. Plotting the resulting points in the order ofincreasing values of t traces the curve in a specific direction. This is called theorientation of the curve.Example 1Sketching a CurveSketch the curve given by the parametric equationsx t2 4andty ,2 2 t 3.SolutionUsing values of t in the interval, the parametric equations yield the points x, y shown in the table.ty6t2 x y t2424t 3t 2t 1xy 20 1 1 3 1 20 401 31 2201353 2xt 0t 12t 2 2 4FIGURE46By plotting these points in the order of increasing t, you obtain the curve Cshown in Figure 10.51. Note that the arrows on the curve indicate its orientationas t increases from 2 to 3. So, if a particle were moving on this curve, it would3start at 0, 1 and then move along the curve to the point 5, 2 . 2 t 3Now try Exercises 1(a) and (b).10.51y64t 122Note that the graph shown in Figure 10.51 does not define y as a function ofx. This points out one benefit of parametric equations—they can be used torepresent graphs that are more general than graphs of functions.It often happens that two different sets of parametric equations have thesame graph. For example, the set of parametric equationsx 4t 2 4y tt 23t 1xt 02t 12 2 t 1 4FIGURE10.5246 1 t 23x 4t 2 4andy t, 1 t 32has the same graph as the set given in Example 1. However, by comparing thevalues of t in Figures 10.51 and 10.52, you see that this second graph is tracedout more rapidly (considering t as time) than the first graph. So, in applications,different parametric representations can be used to represent various speeds atwhich objects travel along a given path.

333202 1006.qxd12/8/059:05 AMPage 773Section 10.6Parametric Equations773Eliminating the ParameterExample 1 uses simple point plotting to sketch the curve. This tedious processcan sometimes be simplified by finding a rectangular equation (in x and y) thathas the same graph. This process is called eliminating the parameter.Parametricequationsx t2 4y t 2Solve fort in oneequation.Substitutein otherequation.Rectangularequationt 2yx 2y 2 4x 4y 2 4Now you can recognize that the equation x 4y 2 4 represents a parabola witha horizontal axis and vertex 4, 0 .When converting equations from parametric to rectangular form, you mayneed to alter the domain of the rectangular equation so that its graph matches thegraph of the parametric equations. Such a situation is demonstrated in Example 2.Emphasize that converting equationsfrom parametric to rectangular form isprimarily an aid in graphing.Example 2ExplorationMost graphing utilities have aparametric mode. If yours does,enter the parametric equationsfrom Example 2. Over whatvalues should you let t vary toobtain the graph shown inFigure 10.53?Eliminating the ParameterSketch the curve represented by the equationsx 1y and t 1tt 1by eliminating the parameter and adjusting the domain of the resulting rectangularequation.SolutionSolving for t in the equation for x producesx x2 1t 1which implies thatParametric equations:1 , y tt 1t 1x 1 t 1t 1 x2.x2yNow, substituting in the equation for y, you obtain the rectangular equation1t 3t 0 2 11 1 2 3FIGURE10.53t 0.75x2 1 x 2 1 x22txx2x2y 1 x 2. t 1 1 x 2 1 x2x2 1 1x2x2 From this rectangular equation, you can recognize that the curve is a parabolathat opens downward and has its vertex at 0, 1 . Also, this rectangular equationis defined for all values of x, but from the parametric equation for x you can seethat the curve is defined only when t 1. This implies that you should restrictthe domain of x to positive values, as shown in Figure 10.53.Now try Exercise 1(c).

333202 1006.qxd77412/8/059:05 AMChapter 10Page 774Topics in Analytic GeometryIt is not necessary for the parameter in a set of parametric equations torepresent time. The next example uses an angle as the parameter.To eliminate the parameter inequations involving trigonometric functions, try using theidentitiesExample 3Eliminating an Angle ParameterSketch the curve represented byx 3 cos sin2 cos2 1sec2 tan2 1y 4 sin ,and0 2 by eliminating the parameter.Solutionas shown in Example 3.Begin by solving for cos and sin in the equations.ycos θ π22 4 1θ 0124 3x 3 cos θy 4 sin θFIGUREy4Solve for cos and sin . 3x 4y 2x2Pythagorean identity 1Substitutex2y2 1916 2θ 3π2sin cos2 sin2 11 2 1andUse the identity sin2 cos2 1 to form an equation involving only x and y.3θ πx3xyfor cos and for sin .34Rectangular equationFrom this rectangular equation, you can see that the graph is an ellipse centeredat 0, 0 , with vertices 0, 4 and 0, 4 and minor axis of length 2b 6, asshown in Figure 10.54. Note that the elliptic curve is traced out counterclockwiseas varies from 0 to 2 .Now try Exercise 13.10.54In Examples 2 and 3, it is important to realize that eliminating the parameter is primarily an aid to curve sketching. If the parametric equations representthe path of a moving object, the graph alone is not sufficient to describe theobject’s motion. You still need the parametric equations to tell you the position,direction, and speed at a given time.Finding Parametric Equations for a GraphYou have been studying techniques for sketching the graph represented by a setof parametric equations. Now consider the reverse problem—that is, how canyou find a set of parametric equations for a given graph or a given physicaldescription? From the discussion following Example 1, you know that such arepresentation is not unique. That is, the equationsx 4t 2 4andy t, 1 t 32produced the same graph as the equationsx t2 4andty , 2 t 3.2This is further demonstrated in Example 4.

333202 1006.qxd12/8/059:05 AMPage 775Section 10.6x 1 ty 2t t 2Example 4yt 1x2FIGUREa. t xb. t 1 xa. Letting t x, you obtain the parametric equationsandx ty 1 x 2 1 t 2. 2 3Finding Parametric Equations for a GraphSolution 1t 3775Find a set of parametric equations to represent the graph of y 1 x 2, usingthe following parameters.t 0t 2 2Parametric Equationst 110.55b. Letting t 1 x, you obtain the parametric equationsx 1 tandy 1 x2 1 1 t 2 2t t 2.In Figure 10.55, note how the resulting curve is oriented by the increasingvalues of t. For part (a), the curve would have the opposite orientation.Now try Exercise 37.Parametric Equations for a CycloidExample 5Point out that a single rectangularequation can have many differentparametric representations. To reinforcethis, demonstrate along with parts (a)and (b) of Example 4 the parametricequation representations of the graphof y 1 x 2 using the parameterst 2x and t 2 3x . A graphingutility can be a helpful tool indemonstrating that each of theserepresentations yields the same graph. representsIn Example 5, PDthe arc of the circle betweenpoints P and D.Describe the cycloid traced out by a point P on the circumference of a circle ofradius a as the circle rolls along a straight line in a plane.SolutionAs the parameter, let be the measure of the circle’s rotation, and let the pointP x, y begin at the origin. When 0, P is at the origin; when , P isat a maximum point a, 2a ; and when 2 , P is back on the x-axis at 2 a, 0 . From Figure 10.56, you can see that APC 180 . So, you haveAC BD aaAPcos cos 180 cos APC asin sin 180 sin APC which implies that AP a cos and BD a sin . Because the circle rolls a . Furthermore, because BA along the x-axis, you know that OD PDDC a, you havex OD BD a a sin (π a, 2a)P (x, y)Use a graphing utility in parametricmode to obtain a graph similarto Figure 10.56 by graphing thefollowing equations.2aaCAOCycloid:x a(θ sin θ), y a(1 cos θ )(3π a, 2a)θBD πa(2π a, 0)X1T T sin TY1T 1 cos Ty BA AP a a cos .So, the parametric equations are x a sin and y a 1 cos .yTe c h n o l o g yandFIGURE10.56Now try Exercise 63.3π a(4π a, 0)x

333202 1006.qxd77612/8/05Chapter 1011:08 AMPage 776Topics in Analytic Geometry10.6 ExercisesVOCABULARY CHECK: Fill in the blanks.1. If f and g are continuous functions of t on an interval I, the set of ordered pairs f t , g t is aC. The equations x f t and y g t are equations for C, and t is the .2. The of a curve is the direction in which the curve is traced out for increasing values of the parameter.3. The process of converting a set of parametric equations to a corresponding rectangular equation is calledthe .PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.1. Consider the parametric equations x t and y 3 t.(a) Create a table of x- and y-values using t 0, 1, 2, 3,and 4.(b) Plot the points x, y generated in part (a), and sketch agraph of the parametric equations.(c) Find the rectangular equation by eliminating theparameter. Sketch its graph. How do the graphs differ?2. Consider the parametric equations x 4 cos 2 andy 2 sin .(a) Create a table of x- and y-values using 2, 4, 0, 4, and 2.(b) Plot the points x, y generated in part (a), and sketch agraph of the parametric equations.(c) Find the rectangular equation by eliminating theparameter. Sketch its graph. How do the graphs differ?In Exercises 3–22, (a) sketch the curve represented by theparametric equations (indicate the orientation of the curve)and (b) eliminate the parameter and write the correspondingrectangular equation whose graph represents the curve.Adjust the domain of the resulting rectangular equation ifnecessary.3. x 3t 34. x 3 2ty 2t 1y 2 3t15. x 4 ty t6. x t7. x t 210. x t 1ty t 1ty t 111. x 2 t 1 y t 2 y 1 sin 19. x e ty 2 3 sin 20. x e2ty e3t21. x y et22. x ln 2tt3y 2t 2y 3 ln tIn Exercises 23 and 24, determine how the plane curves differ from each other.(b) x cos 23. (a) x ty 2 cos 1y 2t 1(c) x y (d) x ete t2e t(b) x t 224. (a) x ty t2y 2et 1 1y t4 1 1(c) x sin t(d) x ety sin t 12y e2t 1In Exercises 25 –28, eliminate the parameter and obtain thestandard form of the rectangular equation.27. Ellipse: x h a cos , y k b sin 28. Hyperbola: x h a sec , y k b tan 12. x t 1y t 2y 2 sin 2 18. x 4 2 cos 26. Circle: x h r cos , y k r sin y 1 t9. x t 1y 2 cos 2 17. x 4 2 cos x x1 t x 2 x1 , y y1 t y2 y1 8. x ty t216. x cos 25. Line through x1, y1 and x2, y2 :y t3215. x 4 sin 2 13. x 3 cos 14. x 2 cos y 3 sin y 3 sin In Exercises 29–36, use the results of Exercises 25–28 tofind a set of parametric equations for the line or conic.29. Line: passes through 0, 0 and 6, 3 30. Line: passes through 2, 3 and 6, 3 31. Circle: center: 3, 2 ; radius: 4

333202 1006.qxd12/8/059:05 AMPage 777Section 10.632. Circle: center: 3, 2 ; radius: 5Parametric Equations77753. Lissajous curve: x 2 cos , y sin 2 33. Ellipse: vertices: 4, 0 ; foci: 3, 0 54. Evolute of ellipse: x 4 cos3 , y 6 sin3 34. Ellipse: vertices: 4, 7 , 4, 3 ;155. Involute of circle: x 2 cos sin foci: (4, 5 , 4, 1 1y 2 sin cos 35. Hyperbola: vertices: 4, 0 ; foci: 5, 0 156. Serpentine curve: x 2 cot , y 4 sin cos 36. Hyperbola: vertices: 2, 0 ; foci: 4, 0 In Exercises 37– 44, find a set of parametric equations forthe rectangular equation using (a) t x and (b) t 2 x.Projectile Motion A projectile is launched at a height of hfeet above the ground at an angle of with the horizontal.The initial velocity is v0 feet per second and the path of theprojectile is modeled by the parametric equations37. y 3x 238. x 3y 239. y x 240. y x3x v0 cos t41. y x 2 142. y 2 x143. y x144. y 2xIn Exercises 57 and 58, use a graphing utility to graph thepaths of a projectile launched from ground level at eachvalue of and v0. For each case, use the graph to approximate the maximum height and the range of the projectile.In Exercises 45–52, use a graphing utility to graph the curverepresented by the parametric equations.and57. (a) 60 ,y h v0 sin t 16t 2.v0 88 feet per second(b) 60 ,v0 132 feet per second45. Cycloid: x 4 sin , y 4 1 cos (c) 45 ,v0 88 feet per second46. Cycloid: x sin , y 1 cos (d) 45 ,v0 132 feet per second3258. (a) 15 ,3247. Prolate cycloid: x sin , y 1 cos 48. Prolate cycloid: x 2 4 sin , y 2 4 cos 49. Hypocycloid: x 3 cos , y 33sin3 v0 60 feet per second(b) 15 ,v0 100 feet per second(c) 30 ,v0 60 feet per second(d) 30 , v0 100 feet per second50. Curtate cycloid: x 8 4 sin , y 8 4 cos 51. Witch of Agnesi: x 2 cot , y 2 sin2 52. Folium of Descartes: x 3t3t 2, y 31 t1 t3Model ItIn Exercises 53–56, match the parametric equations withthe correct graph and describe the domain and range.[The graphs are labeled (a), (b), (c), and (d).]y(a)y(b)2211 2 1x1 1 12x1 1y(d)y55 5x 42 47 ft408 ftNot drawn to scale(a) Write a set of parametric equations that model thepath of the baseball.4x 5θ3 ft 2(c)59. Sports The center field fence in Yankee Stadium is7 feet high and 408 feet from home plate. A baseball ishit at a point 3 feet above the ground. It leaves the batat an angle of degrees with the horizontal at a speedof 100 miles per hour (see figure).(b) Use a graphing utility to graph the path of thebaseball when 15 . Is the hit a home run?(c) Use a graphing utility to graph the path of thebaseball when 23 . Is the hit a home run?(d) Find the minimum angle required for the hit to bea home run.

333202 1006.qxd77812/8/059:05 AMChapter 10Page 778Topics in Analytic Geometry60. Sports An archer releases an arrow from a bow at a point5 feet above the ground. The arrow leaves the bow at anangle of 10 with the horizontal and at an initial speed of240 feet per second.(a) Write a set of parametric equations that model the pathof the arrow.64. Epicycloid A circle of radius one unit rolls around theoutside of a circle of radius two units without slipping. Thecurve traced by a point on the circumference of the smaller circle is called an epicycloid (see figure). Use the angle shown in the figure to find a set of parametric equationsfor the curve.y(b) Assuming the ground is level, find the distance thearrow travels before it hits the ground. (Ignore airresistance.)4(c) Use a graphing utility to graph the path of the arrowand approximate its maximum height.3(d) Find the total time the arrow is in the air.61. Projectile Motion Eliminate the parameter t from theparametric equationsx v0 cos tandy h v0 sin t 16t2for the motion of a projectile to show that the rectangularequation isy 16sec 2 1θ(x, y)13x4Synthesisx 2 tan x h.True or False? In Exercises 65 and 66, determine whetherthe statement is true or false. Justify your answer.62. Path of a Projectile The path of a projectile is given bythe rectangular equationy 7 x 0.02x 2.65. The two sets of parametric equations x t,y t 2 1 and x 3t, y 9t 2 1 have the samerectangular equation.(a) Use the result of Exercise 61 to find h, v0, and . Findthe parametric equations of the path.66. The graph of the parametric equations x t 2 and y t 2 isthe line y x.(b) Use a graphing utility to graph the rectangular equationfor the path of the projectile. Confirm your answerin part (a) by sketching the curve represented by theparametric equations.67. Writing Write a short paragraph explaining whyparametric equations are useful.v02(c) Use a graphing utility to approximate the maximumheight of the projectile and its range.63. Curtate Cycloid A wheel of radius a units rolls along astraight line without slipping. The curve traced by a pointP that is b units from the center b a is called a curtatecycloid (see figure). Use the angle shown in the figure tofind a set of parametric equations for the curve.68. Writing Explain the process of sketching a plane curvegiven by parametric equations. What is meant by theorientation of the curve?Skills ReviewIn Exercises 69–72, solve the system of equations.69.5x 7y 11 3x y 1370. 3x 5y 94x 2y 1471.3a 2b c 82a b 3c 3a 3b 9c 1672.y(π a, a b)2aPbθ(0, a b)aπa2π ax5u 7v 9w 4u 2v 3w 78u 2v w 20In Exercises 73–76, find the reference angle , and sketch and in standard position.73. 105 74. 230 2 75. 376. 5 6

Parametric equations are useful for modeling the path of an object. For instance, in Exercise 59 on page 777, you will use a set of parametric equations to model the path of a baseball. Parametric Equations Jed Jacobsohn/Getty Images 10.6 Rectangular equation: yx x 2 72 (36, 18) (0, 0) (72, 0) t t t 0 Parametric equations: x 24 2t .

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