1. Graphing Parametric Equations:

2. Graphing Parametric Equations:
Suppose that the system of parametric equations given below represents the path that a figure-skater carves on the ice:

3. Graphing Parametric Equations:
Suppose that the system of parametric equations given below represents the path that a figure-skater carves on the ice:
Let’s sketch a graph of the figure-skater’s movement. We can start by constructing a table of values so that we can find ordered pairs on the graph of the system of parametric equations.

4. Graphing Parametric Equations:
To construct a table of values for the system

5. Graphing Parametric Equations:
To construct a table of values for the system
let’s choose values for t

6. Graphing Parametric Equations:
To construct a table of values for the system
let’s choose values for t

7. Graphing Parametric Equations:
To construct a table of values for the system
let’s choose values for t, find corresponding values of x(t)

8. Graphing Parametric Equations:
x(t)
To construct a table of values for the system
let’s choose values for t, find corresponding values of x(t)

9. Graphing Parametric Equations:
x(t)
To construct a table of values for the system
let’s choose values for t, find corresponding values of x(t) and y(t),

10. Graphing Parametric Equations:
x(t)
y(t)
To construct a table of values for the system
let’s choose values for t, find corresponding values of x(t) and y(t),

11. Graphing Parametric Equations:
x(t)
y(t)
To construct a table of values for the system
let’s choose values for t, find corresponding values of x(t) and y(t), and then plot ordered pairs (x(t), y(t))

12. Graphing Parametric Equations:
x(t)
y(t)
(x(t), y(t) )
To construct a table of values for the system
let’s choose values for t, find corresponding values of x(t) and y(t), and then plot ordered pairs (x(t), y(t))

13. Graphing Parametric Equations:
x(t)
y(t)
(x(t), y(t) )
To construct a table of values for the system
let’s choose values for t, find corresponding values of x(t) and y(t), and then plot ordered pairs (x(t), y(t)) on a coordinate plane.

14. Graphing Parametric Equations:
x(t)
y(t)
(x(t), y(t) )
To construct a table of values for the system
let’s choose values for t, find corresponding values of x(t) and y(t), and then plot ordered pairs (x(t), y(t)) on a coordinate plane.

15. Graphing Parametric Equations:
x(t)
y(t)
(x(t), y(t) )

16. Graphing Parametric Equations:
x(t)
y(t)
(x(t), y(t) )
Let’s start with t = 0.

17. Graphing Parametric Equations:
x(t)
y(t)
(x(t), y(t) )
Let’s start with t = 0.

18. Graphing Parametric Equations:
x(t)
y(t)
(x(t), y(t) )
Let’s start with t = 0. Then

19. Graphing Parametric Equations:
x(t)
y(t)
(x(t), y(t) )
Let’s start with t = 0. Then

20. Graphing Parametric Equations:
x(t)
y(t)
(x(t), y(t) )
Let’s start with t = 0. Then
and

21. Graphing Parametric Equations:
x(t)
y(t) 8
(x(t), y(t) )
Let’s start with t = 0. Then
and

22. Graphing Parametric Equations:
x(t)
y(t) 8
(x(t), y(t) )
Let’s start with t = 0. Then
and
So (0, 8) is on the graph of this system.

23. Graphing Parametric Equations:
x(t)
y(t) 8
(x(t), y(t) )
(0, 8)
Let’s start with t = 0. Then
and
So (0, 8) is on the graph of this system.

24. Graphing Parametric Equations:
x(t)
y(t) 8
(x(t), y(t) )
(0, 8)
Let’s start with t = 0. Then
and
So (0, 8) is on the graph of this system. Let’s plot this point on our coordinate plane.

25. Graphing Parametric Equations:
x(t)
y(t) 8
(x(t), y(t) )
(0, 8)

26. Graphing Parametric Equations:
x(t)
y(t) 8
(x(t), y(t) )
(0, 8)
Now let’s take t = 1.

27. Graphing Parametric Equations:1
x(t)
y(t) 8
(x(t), y(t) )
(0, 8)
Now let’s take t = 1.

28. Graphing Parametric Equations:1
x(t)
y(t) 8
(x(t), y(t) )
(0, 8)
Now let’s take t = 1. Then

29. Graphing Parametric Equations:1
x(t) 4
y(t) 8
(x(t), y(t) )
(0, 8)
Now let’s take t = 1. Then

30. Graphing Parametric Equations:1
x(t) 4
y(t) 8
(x(t), y(t) )
(0, 8)
Now let’s take t = 1. Then
and

31. Graphing Parametric Equations:1
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
Now let’s take t = 1. Then
and

32. Graphing Parametric Equations:1
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
Now let’s take t = 1. Then
and
So is on the graph of this system.
(4, 5.7)

33. Graphing Parametric Equations:1
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(4, 5.7)
Now let’s take t = 1. Then
and
So is on the graph of this system.
(4, 5.7)

34. Graphing Parametric Equations:1
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(4, 5.7)
Now let’s take t = 1. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane
(4, 5.7)

35. Graphing Parametric Equations:1
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(4, 5.7)
Now let’s take t = 1. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane and connect our points.
(4, 5.7)

36. Graphing Parametric Equations:1
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(4, 5.7)

37. Graphing Parametric Equations:1
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(4, 5.7)
Now let’s take t = 2.

38. Graphing Parametric Equations:1 2
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(4, 5.7)
Now let’s take t = 2.

39. Graphing Parametric Equations:1 2
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(4, 5.7)
Now let’s take t = 2. Then

40. Graphing Parametric Equations:1 2
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(4, 5.7)
Now let’s take t = 2. Then

41. Graphing Parametric Equations:1 2
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(4, 5.7)
Now let’s take t = 2. Then
and

42. Graphing Parametric Equations:1 2
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(4, 5.7)
Now let’s take t = 2. Then
and

43. Graphing Parametric Equations:1 2
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(4, 5.7)
Now let’s take t = 2. Then
and
So is on the graph of this system.
(0, 0)

44. Graphing Parametric Equations:1 2
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
Now let’s take t = 2. Then
and
So is on the graph of this system.
(0, 0)

45. Graphing Parametric Equations:1 2
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
Now let’s take t = 2. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane
(0, 0)

46. Graphing Parametric Equations:1 2
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
Now let’s take t = 2. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane and connect our points.
(0, 0)

47. Graphing Parametric Equations:1 2
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)

48. Graphing Parametric Equations:1 2
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
Now let’s take t = 3.

49. Graphing Parametric Equations:1 2 3
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
Now let’s take t = 3.

50. Graphing Parametric Equations:1 2 3
x(t) 4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
Now let’s take t = 3. Then

51. Graphing Parametric Equations:1 2 3
x(t) 4 -4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
Now let’s take t = 3. Then

52. Graphing Parametric Equations:1 2 3
x(t) 4 -4
y(t) 8
5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
Now let’s take t = 3. Then
and

53. Graphing Parametric Equations:1 2 3
x(t) 4 -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
Now let’s take t = 3. Then
and

54. Graphing Parametric Equations:1 2 3
x(t) 4 -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
Now let’s take t = 3. Then
and
So is on the graph of this system.
(-4, -5.7)

55. Graphing Parametric Equations:1 2 3
x(t) 4 -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
(-4, -5.7)
Now let’s take t = 3. Then
and
So is on the graph of this system.
(-4, -5.7)

56. Graphing Parametric Equations:1 2 3
x(t) 4 -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
(-4, -5.7)
Now let’s take t = 3. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane
(-4, -5.7)

57. Graphing Parametric Equations:1 2 3
x(t) 4 -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
(-4, -5.7)
Now let’s take t = 3. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane and connect our points.
(-4, -5.7)

58. Graphing Parametric Equations:1 2 3
x(t) 4 -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
(-4, -5.7)

59. Graphing Parametric Equations:1 2 3
x(t) 4 -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
(-4, -5.7)
Now let’s take t = 4.

60. Graphing Parametric Equations:1 2 3 4
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
(-4, -5.7)
Now let’s take t = 4.

61. Graphing Parametric Equations:1 2 3 4
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
(-4, -5.7)
Now let’s take t = 4. Then

62. Graphing Parametric Equations:1 2 3 4
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
(-4, -5.7)
Now let’s take t = 4. Then

63. Graphing Parametric Equations:1 2 3 4
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(4, 5.7)
(-4, -5.7)
Now let’s take t = 4. Then
and

64. Graphing Parametric Equations:1 2 3 4
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0) -8
(4, 5.7)
(-4, -5.7)
Now let’s take t = 4. Then
and

65. Graphing Parametric Equations:1 2 3 4
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0) -8
(4, 5.7)
(-4, -5.7)
Now let’s take t = 4. Then
and
So is on the graph of this system.
(0, -8)

66. Graphing Parametric Equations:1 2 3 4
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
Now let’s take t = 4. Then
and
So is on the graph of this system.
(0, -8)

67. Graphing Parametric Equations:1 2 3 4
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
Now let’s take t = 4. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane
(0, -8)

68. Graphing Parametric Equations:1 2 3 4
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
Now let’s take t = 4. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane and connect our points.
(0, -8)

69. Graphing Parametric Equations:1 2 3 4
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)

70. Graphing Parametric Equations:1 2 3 4
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
Now let’s take t = 5.

71. Graphing Parametric Equations:1 2 3 4 5
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
Now let’s take t = 5.

72. Graphing Parametric Equations:1 2 3 4 5
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
Now let’s take t = 5. Then

73. Graphing Parametric Equations:1 2 3 4 5
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
Now let’s take t = 5. Then

74. Graphing Parametric Equations:1 2 3 4 5
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
Now let’s take t = 5. Then
and

75. Graphing Parametric Equations:1 2 3 4 5
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
Now let’s take t = 5. Then
and

76. Graphing Parametric Equations:1 2 3 4 5
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
Now let’s take t = 5. Then
and
So is on the graph of this system.
(4, -5.7)

77. Graphing Parametric Equations:1 2 3 4 5
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 5. Then
and
So is on the graph of this system.
(4, -5.7)

78. Graphing Parametric Equations:1 2 3 4 5
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 5. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane
(4, -5.7)

79. Graphing Parametric Equations:1 2 3 4 5
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 5. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane
and connect our points.
(4, -5.7)

80. Graphing Parametric Equations:1 2 3 4 5
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)

81. Graphing Parametric Equations:1 2 3 4 5
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 6.

82. Graphing Parametric Equations:1 2 3 4 5 6
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 6.

83. Graphing Parametric Equations:1 2 3 4 5 6
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 6. Then

84. Graphing Parametric Equations:1 2 3 4 5 6
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 6. Then

85. Graphing Parametric Equations:1 2 3 4 5 6
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 6. Then
and

86. Graphing Parametric Equations:1 2 3 4 5 6
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 6. Then
and

87. Graphing Parametric Equations:1 2 3 4 5 6
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 6. Then
and
So is on the graph of this system.
(0, 0)

88. Graphing Parametric Equations:1 2 3 4 5 6
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 6. Then
and
So is on the graph of this system.
(0, 0)

89. Graphing Parametric Equations:1 2 3 4 5 6
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 6. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane and
connect our points.
(0, 0)

90. Graphing Parametric Equations:1 2 3 4 5 6
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)

91. Graphing Parametric Equations:1 2 3 4 5 6
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 7.

92. Graphing Parametric Equations:1 2 3 4 5 6 7
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 7.

93. Graphing Parametric Equations:1 2 3 4 5 6 7
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 7. Then

94. Graphing Parametric Equations:1 2 3 4 5 6 7
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 7. Then

95. Graphing Parametric Equations:1 2 3 4 5 6 7
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 7. Then
and

96. Graphing Parametric Equations:1 2 3 4 5 6 7
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 7. Then
and

97. Graphing Parametric Equations:1 2 3 4 5 6 7
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
Now let’s take t = 7. Then
and
So is on the graph of this system.
(-4, 5.7)

98. Graphing Parametric Equations:1 2 3 4 5 6 7
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
(-4, 5.7)
Now let’s take t = 7. Then
and
So is on the graph of this system.
(-4, 5.7)

99. Graphing Parametric Equations:1 2 3 4 5 6 7
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
(-4, 5.7)
Now let’s take t = 7. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane
(-4, 5.7)

100. Graphing Parametric Equations:1 2 3 4 5 6 7
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
(-4, 5.7)
Now let’s take t = 7. Then
and
So is on the graph of this system. Let’s plot this point on our coordinate plane and connect our points.
(-4, 5.7)

101. Graphing Parametric Equations:1 2 3 4 5 6 7
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
(-4, 5.7)

102. Graphing Parametric Equations:1 2 3 4 5 6 7
x(t) -4
y(t) 8
5.7
-5.7
(x(t), y(t) )
(0, 8)
(0, 0)
(0, -8) -8
(4, 5.7)
(-4, -5.7)
(4, -5.7)
(-4, 5.7)
Finally, we can finish our graph.
We can now see that the figure skater carved a figure-eight on the ice.