1. 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

2. CH 9KM & PP AIM22 Sections of a Cone

3. CH 9KM & PP AIM23 Sections of a Cone... continued

4. CH 9KM & PP AIM24 Degenerate Conic Sections

5. 9/22/2015Math 120 - KM5 9.1

6. 9/22/2015Math 120 - KM6 The Parabola 9.1

7. 9/22/2015Math 120 - KM7 A Parabolic Reflector For a Microphone Can You Hear a Pin Drop? 9.1

8. 9/22/2015Math 120 - KM8 A Parabolic Archway Architectural Parabola 9.1

9. 9/22/2015Math 120 - KM9 A Parabolic Headlight Shine Your Light Forward 9.1

10. 9/22/2015Math 120 - KM10 Parabolic Shadows 9.1

11. 9/22/2015Math 120 - KM11 y = ax 2 + bx + c a > 0 a < 0 x = ay 2 + by + c a > 0 a < 0 9.1 The Basic Ideas 9.1

12. 9/22/2015Math 120 - KM12 {-4,2} {5/3} {17.5327} {3} Vertex: (-2, -3) Opens upwards (narrow) Axis of symmetry: x = -2 y -intercept: (0,5) 9.1 Ex 1: y = 2x 2 + 8x + 5 9.1

13. 9/22/2015Math 120 - KM13 {-4,2} {5/3} {17.5327} {3} Vertex: (-2, -3) Opens upwards (narrow) Axis of symmetry: x = -2 y -intercept: (0,5) 9.1 Ex 1: y = 2x 2 + 8x + 5 alternate method 9.1

14. 9/22/2015Math 120 - KM14 {-4,2} {5/3} {17.5327} {3} Vertex: (1, 4) Opens downward (narrow) Axis of symmetry: x = 1 y -intercept: (0,-2) 9.1 Ex 2: y = -6x 2 + 12x - 2 9.1

15. 9/22/2015Math 120 - KM15 Vertex: (1, 4) Opens downward (narrow) Axis of symmetry: x = 1 y -intercept: (0-2) 9.1 Ex 2: y = -6x 2 + 12x – 2 alternate method 9.1

16. 9/22/2015Math 120 - KM16 Vertex: (-3, 2) Opens to the right (narrow) Axis of symmetry: y = 2 x – intercept: (5, 0) 9.1 Ex 3: x = 2y 2 – 8y + 5 9.1

17. 9/22/2015Math 120 - KM17 Vertex: (-3, 2) Opens to the right (narrow) Axis of symmetry: y = 2 x – intercept: (5, 0) 9.1 Ex 3: x = 2y 2 – 8y + 5 alternate method 9.1

18. 9/22/2015Math 120 - KM18 Vertex: (-1, -1) Opens to the left (narrow) Axis of symmetry: y = -1 x – intercept: (-3, 0) 9.1 Ex 4: x = -2y 2 – 4y - 3 9.1

19. 9/22/2015Math 120 - KM19 Vertex: (-1, -1) Opens to the left (narrow) Axis of symmetry: y = -1 x – intercept: (-3, 0) 9.1 Ex 4: x = -2y 2 – 4y – 3 alternate method 9.1

20. 9/22/2015Math 120 - KM20 c a b The Distance Formula 9.1

21. 9/22/2015Math 120 - KM21 Determine the distance from P 1 to P 2. P 1 (-2, 3) P 2 (2, 0) P 1 (5, -2) P 2 (-3, -1) 9.1 Distance Formula Examples 9.1

22. 9/22/2015Math 120 - KM22 9.1 MIDPOINT 9.1

23. 9/22/2015Math 120 - KM23 AVERAGE ! 9.1 Average the Coordinates! 9.1

24. 9/22/2015Math 120 - KM24 Determine the midpoint of P 1 P 2. P 1 (-2, 3) P 2 (2, 0) P 1 (5, -2) P 2 (-3, -1) 9.1 Midpoint Examples 9.1

25. 9/22/2015 9:03 PM krm 11.225 With a COMPASS How do I make a circle ? 9.1 Circles 9.1

26. 9/22/2015 9:03 PM krm 11.226 The set of all points in a plane that are at a fixed distance, r, called the radius from a fixed point, (h, k), called the center. 9.1 Circle: Center (h,k) Radius r 9.1

27. 9/22/2015Math 120 - KM27 9.1 x 2 + y 2 = 1 9.1

28. 9/22/2015Math 120 - KM28 9.1 (x + 2) 2 + (y – 4) 2 = 3 2 9.1

29. 9/22/2015Math 120 - KM29 9.1 x 2 + (y + 4) 2 = 25 9.1

30. 9/22/2015 9:03 PM krm 11.230 9.1 Write the equation of the circle with radius 7 and center (-5, 8). 9.1

31. 9/22/2015 9:03 PM krm 11.231 Look for ax 2 + ay 2 How do I know it’s a circle ? The Equation of a Circle 9.1

32. 9/22/2015 9:03 PM krm 11.232 Write the equation of the circle in standard form and sketch the graph: x 2 + y 2 - 6x + 10y + 25 = 0 Circle: Standard Form 9.1

33. 9/22/2015Math 120 - KM33 9.2 The Ellipse 9.2

34. 9/22/2015Math 120 - KM34 x-intercepts (+ a, 0) y-intercepts (0, + b) 9.2 Ellipse (it fits in a box!) 9.2

35. 9/22/2015Math 120 - KM35 9.2 Example: Horizontal Major Axis 9.2

36. 9/22/2015Math 120 - KM36 9.2 Example: Vertical Major Axis 9.2

37. 9/22/2015Math 120 - KM37 9.2 Example: center not at the origin 9.2

38. 9/22/2015Math 120 - KM38 9.2 Example: Put in Standard Form First 9.2

39. 9/22/2015Math 120 - KM39 9.2 Example continued: Put in Standard Form First 9.2

40. 9/22/2015Math 120 - KM40 9.3 The Hyperbola it fits outside the box 9.3

41. 9/22/2015Math 120 - KM41 9.3 The Hyperbola STANDARD FORM 9.3

42. 9/22/2015Math 120 - KM42 1.Fundamental Rectangle 2.Asymptotes 3.Vertices (if x 2 – y 2 …) 4.Sketch 9.3 Hyperbola: x 2 is first 9.3

43. 9/22/2015Math 120 - KM43 9.3 Example x 2 is first 9.3

44. 9/22/2015Math 120 - KM44 1.Fundamental Rectangle 2.Asymptotes 3.Vertices (if y 2 – x 2 …) 4.Sketch 9.3 Hyperbola: y 2 is first 9.3

45. 9/22/2015Math 120 - KM45 9.3 Example y 2 is first 9.3

46. 9/22/2015Math 120 - KM46 9.3 The Hyperbola NONSTANDARD FORM 9.3

47. 9/22/2015Math 120 - KM47 9.3 The Hyperbola NONSTANDARD FORM Example 1 xy -4 -2 -4 0N 14 22 41 9.3

48. 9/22/2015Math 120 - KM48 9.3 The Hyperbola NONSTANDARD FORM Example 2 xy -41 -22 4 0N 1-4 2-2 4 9.3

49. 9/22/2015Math 120 - KM49 “Conic sections are among the oldest curves, and is an oldest math subject studied systematically and thoroughly. The conics seems to have been discovered by Menaechmus (a Greek, c.375-325 BC), tutor to Alexander the Great. They were conceived in an attempt to solve the three famous problems of trisecting the angle, duplicating the cube, and squaring the circle. The conics were first defined as the intersection of: a right circular cone of varying vertex angle; a plane perpendicular to an element of the cone. (An element of a cone is any line that makes up the cone) Depending the angle is less than, equal to, or greater than 90 degrees, we get ellipse, parabola, or hyperbola respectively. Appollonius (estimated c. 262-190 BC) (known as The Great Geometer) consolidated and extended previous results of conics into a monograph Conic Sections, consisting of eight books with 487 propositions. Quote from Morris Kline: "As an achievement it [Appollonius' Conic Sections] is so monumental that it practically closed the subject to later thinkers, at least from the purely geometrical standpoint." Book VIII of Conic Sections is lost to us. Appollonius' Conic Sections and Euclid's Elements may represent the quintessence of Greek mathematics. Appolloniuswas the first to base the theory of all three conics on sections of one circular cone, right or oblique. He is also the one to give the name ellipse, parabola, and hyperbola. A brief explanation of the naming can be found in Howard Eves, An Introduction to the History of Math. 6th ed. page 172. In Renaissance, Kepler's law of planetary motion, Descarte and Fermat's coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, Pascal pushed conics to a high level. Many later mathematicians have also made contribution to conics, espcially in the development of projective geometry where conics are fundamental objects as circles in Greek geometry. Among the contributors, we may find Newton, Dandelin, Gergonne, Poncelet, Brianchon, Dupin, Chasles, and Steiner. Conic sections is a rich classic topic that has spurred many developments in the history of mathematics.” From the website: http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html” Conics...2300+ years old?

50. 9/22/2015Math 120 - KM50 9.4

51. 9/22/2015Math 120 - KM51  Think of the Possibilities! 9.4

52. 9/22/2015Math 120 - KM52 Where will they meet? 9.4

53. 9/22/2015Math 120 - KM53 Where will they meet - exactly? 9.4

54. 9/22/2015Math 120 - KM54 Where will they meet - exactly? 9.4

55. 9/22/2015Math 120 - KM55 Where will they meet - exactly? 9.4

56. 9/22/2015Math 120 - KM56 How about a really tough one? 9.4

57. 9/22/2015Math 120 - KM57 How about a really tough one? Continued... 9.4

58. 9/22/2015Math 120 - KM58 How about a really tough one? Continued... 9.4

59. 9/22/2015Math 120 - KM59 That’s All for Now!