1. 9.5 Parametric Equations 2015 Calculator

2. Ships in the Fog

3. Objective To evaluate sets of parametric equations for given values of the parameter. To sketch curves that are represented by sets of parametric equations To rewrite sets of parametric equations as single rectangular equations by eliminating the parameter.

4. Suppose you were running around an elliptical shaped track. You might be following the elliptical path modeled by the equation: This equation only shows you where you are, it doesn’t show you when you are at a given point (x, y) on the track. To determine this time, we introduce a third variable t, called a parameter. We can write both x and y as functions of t to obtain parametric equations. Parametric Equations

5. Definition of a Plane Curve If f and g are continuous functions of t on an interval, the set of ordered pairs (f(t), g(t)) graphs out a plane curve. The equations x = f(t) and y = g(t) are parametric equations, and t is the parameter. Any parameter can be used, but t is commonly used as a parameter to represent time.

6. Sketching a Plane Curve When sketching a curve represented by a pair of parametric equations, you still plot points in the xy-plane. Each set of coordinates (x, y) is determined from a value chosen for the parameter t.

7. Example: Sketching a Plane Curve Example: Sketch the curve given by x = t + 2 and y = t 2, – 3  t  3. t– 3– 2– 10123 x 012345 y9410149 y x -4 4 4 8 orientation of the curve

8. Graphing Utility: Sketching a Curve Plane Graphing Utility: Sketch the curve given by x = t + 2 and y = t 2, – 3  t  3. Mode Menu: Set to parametric mode. WindowGraphTable

9. Ex: sketch the curve given by the parametric equations, then eliminate the parameter and find the rectangular equation: for t– 2– 10123 x0– 3– 4– 305 y– 1–.50.511.5 y x -4 4 4 8

10. Example: Finding Parametric Equations Parametric equation for x. Substitute into the original rectangular equation. Example: Find a set of parametric equations to represent the graph of y = 4x – 3. Use the parameter t = x. x = t y = 4t – 3 x y -4 4 4 8 t– 2– 1012 x–2–3012 y– 11–7–315

11. Homework P.673 1-8 all.

12. Example: Sketching a Plane Curve Example: Sketch the curve given by x = t + 2 and y = t 2, – 3  t  3. t– 3– 2– 10123 x 012345 y9410149 y x -4 4 4 8 orientation of the curve

13. Eliminating the parameter is a process for finding the rectangular equation (in x and y) of a curve represented by parametric equations. Definition: Eliminating the Parameter x = t + 2 y = t 2 Parametric equations t = x – 2 Solve for t in one equation. y = (x –2) 2 Substitute into the second equation. y = (x –2) 2 Equation of a parabola with the vertex at (2, 0)

14. Example: Eliminating the Parameter Solve for t in one equation. Substitute into the second equation. Example: Identify the curve represented by x = 2t and by eliminating the parameter. y x -4 4 4 8

15. Example: Sketching a Plane Curve You try: Find a set of parametric equations to represent the graph of y = 4x – 3. Use the parameter t = 2 – x. t– 2– 1012 x43210 y13951-3 y x -4 4 4 8 x = 2 – t y = 4(2-t )– 3 = 8– 4t – 3 =5– 4t

16. Identify the curve represented by the equations by eliminating the parameter:

17. Eliminate the parameter to identify the curve represented by the parametric equations:

18. Parametric Conics The use of two of the three Pythagorean Trigonometric Identities allow for easy parametric representation on ellipses, hyperbolas, and circles.

19. Pythagorean Identities

20. y x -4 4 4 8 The set of all points satisfying the equation gives the circle with center (h,k). Base = x-h height = y-k Pythagorean Theorem? r

21. Or, thinking algebraically: Compare the standard form of a circle with the Pythagorean Identity:

23. Solving for x and y:

25. Find a set of parametric equations to represent the conic, then graph in parametric mode: Ex:

29. Write and Graph a set of parametric equations for the function: Ex:

33. Horizontal Axis Ellipse: Vertical Axis Ellipse : Horizontal Axis Hyperbola:Vertical Axis Hyperbola:

35. Eliminate the parameter. Describe the graph of the conic. Ex :

36. Eliminate the parameter. Describe the graph of the conic. Ex :

37. You try: Sketch the curve represented by by eliminating the parameter. Describe the graph of the conic.

38. Find a set of parametric equations for the ellipse with vertices and foci. You try :

39. Parametric equations of a line separate the horizontal and vertical components of points on the line. Find parametric equations to represent the line that passes through The x equation describes the horizontal position for some change in the parameter. The y equation describes the vertical position for some change in the parameter.

40. Parametric equations of a line

41. You try: Find the parametric equations to represent the line that passes through.

42. Example: The motion of a projectile at time t (in seconds) is given by the parametric equations: Where x(t) gives the horizontal position of the projectile in feet and y(t) gives the vertical position of the projectile in feet.

43. a. Find the vertical and horizontal position of the projectile when t = 2 x = 50, y = 6 b. At what time will the projectile hit the ground? The ball will hit the ground between t = 2.16 and t = 2.18

44. Homework P.673 9-29 odd, 33-36 all, 37-41 odd.

45. Extras Optional Additional Probs, The parametric equations below represent the hawk and dove populations at time t, where t is measured in years.

46. a. Use your calculator in function mode to graph the hawk and dove populations over time. Dove Hawk

47. b. Find the maximum and minimum values for each population. Hawk minimum 10 maximum 30 Dove minimum 50 maximum 250

48. c. Now using Parametric mode on your calculator, graph the hawk population versus the dove As the hawk population increases, the dove populations decreases, followed by a decrease in hawk population and a decrease in the dove population.

49. d. Using the parametric graph, find the population of hawks and doves after one year. Dove population is 250, hawk population is 20

50. e. When will the population of hawks reach its maximum value and what is that value? Hawk population will be 30 at year 2.

51. Example 9 The complete graph of the parametric equations x = 2cos t and y = 2 sin t is the circle of radius 2 centered at the origin. Find an interval of values for t so that the graph is the given portion of the circle. A) the portion in the first quadrant. (0, π/2) B) the portion above the x-axis. (0, π) C) the portion to the left of the y-axis –(π/2, 3π/2)

52. Example: Ron is on a Ferris wheel of radius 35 ft that turns councterclockwise at the rate of one revolution every 12 seconds. The lowest point of the Ferris wheel is 15 feet above ground level at the point, (0, 15) on a rectangular coordinate system. Find parametric equations for the position of Ron as a function of time t in seconds if the Ferris wheel starts with Ron at the point (35, 50)

55. Example: Al and Betty are on a Ferris wheel. The wheel has a radius of 15 feet and its center is 20 feet above the ground. How high are Al and Betty ath the 3 o’clock position? At the 12 o’clock position? At the 9 o’clock position?

56. Example: A dart is thrown upward with an initial velocity of 58 ft/sec at an angle of elevation of 41°. Find the parametric equations that model the problem situation. Whne will the dart hit the ground? Find the maximum height of the dart. When will this occur?

58. The dart will hit the ground at about 2.51 seconds. The maximum height of the dart is 26.6 feet. This will occur at 1.22 seconds.