A2-level maths UK

1. A2 Mathematics: C4 Core Maths Curves and Tangents Parametric Curves

2. Objectives We will be able to Plot Graphs defined by parametric equations by hand and by calculator Use algebra to eliminate the parameter and find the Cartesian equation of the curve. Find the gradient of the curve for any value of the parameter. Find the equation of the tangent or normal to the curve at any value of the parameter.

3. What is a Parametric Graph ? To plot a graph we could follow a point as it crawls along the curve especially If the point obeys a rule If it gives x and y In terms of time Or other parameter

4. Tracing out a Parametric Graph http://www.flashandmath.com/mathlets/calc/param2d/param_advanced.html

5. Tracing out a Parametric Graph This also shown in your WEC text-book On page 323

6. Parametric Curve examples

7. Parametric Curve examples

8. Parametric Curve examples

9. Parametric Equations for a Curve x=2t, y=15t– 5t²

10. Plotting x and y via parameters

11. Curves defined by parametric equations

12. Parametric Equations for a Curve x = 3cosθ, y = 3sinθ

13. Plotting x and y via parameters

14. Curves defined by parametric equations

15. Plotting Parametric Curves Use Sharp EL9900 calculator Parametric settings on next slide Use Autograph Equation entry via x=2t, y=t^2 Separated by a comma

16. Parametric Settings for EL9900

17. Parametric Entry for EL9900

18. Parametric Displays on the EL9900

19. Cartesian Equation for a Curve We have x as a function of t or θ And yas a function of t or θ We need to eliminate t or θ Leaving only x and y. Methods Eliminate t by substitution and algebra Eliminate θ via trigonometric Identities and algebra

20. Cartesian Equation – Eliminate t x = t2, y = t – t2

21. Cartesian Equation – identities in θ x = 3cosθ, y = sinθ

22. Activity step 1 Use table of values to plot each curve (and/or use your calculator). Match each parameter formula and its curve with correct curve card. Match each curves with its correct Cartesian equation.

23. Parametric Equations for a Curve x = 3cosθ, y = 3sinθ

24. Cartesian Equation for a Curve x2 + y2 = 9

25. Curves defined by parametric equations

26. Parametric Equations for a Curve x=2t, y=15t– 5t²

27. Cartesian Equation for a Curve 4y = 15x– 4.9x2

28. Curves defined by parametric equations

29. Parametric Equations for a Curve x=t²–4, y=t³–4t

30. Cartesian Equation for a Curve y = x√(x+4) y = x(x+4)0.5

31. Curves defined by parametric equations

32. Parametric Equations for a Curve x=sinθ, y=sin2θ

33. Cartesian Equation for a Curve y = 2x√(1-x2) y = 2x(1-x2)0.5

34. Curves defined by parametric equations

35. Parametric Equations for a Curve x=t2, y=t3

36. Cartesian Equation for a Curve y=x√x

37. Curves defined by parametric equations

38. Parametric Equations for a Curve x=t, y=1/t

39. Cartesian Equation for a Curve y = 1/x

40. Curves defined by parametric equations

41. Parametric Equations for a Curve x = 1+ t, y = 2 - t

42. Cartesian Equation for a Curve x + y = 3

43. Curves defined by parametric equations

44. Parametric Equations for a Curve x=(2+3t)/(1+t), y=(3–2t)/(1+t)

45. Parametric Equations for a Curve y=13–5x

46. Curves defined by parametric equations Stops here !

47. Extension: Try these Parameters x= t + 1/t, y= t - 1/t x = 3cosθ, y= sinθ Investigate/Create your own

48. Parametric Equations for a Curve x=………...., y=…….……..

49. Curves defined by parametric equations

50. Tangents to the curve? How do we find dy/dx ? How do we find the equation of the tangent at one particular point on the curve – for example when t=1

51. Parametric Equations for a Curve x = 3cosθ, y = 3sinθ

52. Gradient of Tangents to the Curve We know (why?) that

53. Gradient of Tangents to the Curve x = 3cosθ, y = 3sinθ ...so.....

54. Gradient of Tangents to the Curve Putting it together.......

56. the co-ordinates where it touches the curve.......so.....

57. Equation of one Tangent to Circle

58. Equation of one Tangent to Circle x = 3cosπ/4, y = 3sin π/4 ...so.....

59. Image of one Tangent to the Curve

60. Activity step 2 Use x and y parameter functions, to match dy/dx equation one tangent equation with previous cards

61. Parametric Equations for a Curve x=2t, y=15t– 5t²

62. Gradient of Tangents to the Curve

63. Image of one Tangent to the Curve

64. Equation of one Tangent to the Curve

65. Parametric Equations for a Curve x=t²–4, y=t³–4t

66. Gradient of Tangents to the Curve

67. Image of one Tangent to the Curve

68. Equation of One Tangent to the Curve

69. Parametric Equations for a Curve x=sinθ, y=sin2θ

70. Gradient of Tangents to the Curve

71. Image of Tangent to Curve

72. Equation of One Tangent to the Curve

73. Parametric Equations for a Curve x=t2, y=t3

74. Gradient of Tangents to the Curve

75. Image of Tangent to Curve

76. Equation of one Tangent to the Curve

77. Parametric Equations for a Curve x=t, y=1/t

78. Gradient of Tangents to the Curve

79. Image of Tangent to Curve

80. Equation of one Tangent to the Curve

81. Parametric Equations for a Curve x = 1+ t, y = 2 - t

82. Gradient of Tangents to the Curve

83. Image of Tangent to Curve

84. Equation of one Tangent to the Curve

85. Parametric Equations for a Curve x=(2+3t)/(1+t), y=(3–2t)/(1+t)

86. Gradient of Tangents to the Curve

87. Image of Tangent to Curve

88. Equation of one Tangent to the Curve

89. Parametric Equations for a Curve x=………...., y=…….……..

90. Curves defined by parametric equations

91. Gradient of Tangents to the Curve

92. Equation of one Tangent to the Curve

93. Image of Tangent to Curve