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Chapter 9: Conic Sections

Chapter 9: Conic Sections. 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems. Sections of a Cone. circle. ellipse. parabola. Sections of a Cone ... continued. hyperbola.

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Chapter 9: Conic Sections

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

  2. Sections of a Cone circle ellipse parabola KM & PP AIM2

  3. Sections of a Cone ... continued hyperbola KM & PP AIM2

  4. Degenerate Conic Sections point line Intersecting lines KM & PP AIM2

  5. 9.1 Parabolas and Circles Math 120 - KM

  6. The Parabola 9.1 Math 120 - KM

  7. Can You Hear a Pin Drop? A Parabolic Reflector For a Microphone 9.1 Math 120 - KM

  8. Architectural Parabola A Parabolic Archway 9.1 Math 120 - KM

  9. Shine Your Light Forward A Parabolic Headlight 9.1 Math 120 - KM

  10. Parabolic Shadows 9.1 Math 120 - KM

  11. 9.1 The Basic Ideas 9.1 Math 120 - KM

  12. 9.1 Ex 1: y = 2x2 + 8x + 5 Vertex: (-2, -3) Opens upwards (narrow) Axis of symmetry: x = -2 y -intercept: (0,5) 9.1 {-4,2} {5/3} {17.5327} {3} Math 120 - KM

  13. 9.1 Ex 1: y = 2x2 + 8x + 5alternate method Vertex: (-2, -3) Opens upwards (narrow) Axis of symmetry: x = -2 y -intercept: (0,5) 9.1 {-4,2} {5/3} {17.5327} {3} Math 120 - KM

  14. 9.1 Ex 2: y = -6x2 + 12x - 2 Vertex: (1, 4) Opens downward (narrow) Axis of symmetry: x = 1 y -intercept: (0,-2) 9.1 {-4,2} {5/3} {17.5327} {3} Math 120 - KM

  15. 9.1 Ex 2: y = -6x2 + 12x – 2alternate method Vertex: (1, 4) Opens downward (narrow) Axis of symmetry: x = 1 y -intercept: (0-2) 9.1 Math 120 - KM

  16. 9.1 Ex 3: x = 2y2 – 8y + 5 Vertex: (-3, 2) Opens to the right (narrow) Axis of symmetry: y = 2 x – intercept: (5, 0) 9.1 Math 120 - KM

  17. 9.1 Ex 3: x = 2y2 – 8y + 5alternate method Vertex: (-3, 2) Opens to the right (narrow) Axis of symmetry: y = 2 x – intercept: (5, 0) 9.1 Math 120 - KM

  18. 9.1 Ex 4: x = -2y2 – 4y - 3 Vertex: (-1, -1) Opens to the left (narrow) Axis of symmetry: y = -1 x – intercept: (-3, 0) 9.1 Math 120 - KM

  19. 9.1 Ex 4: x = -2y2 – 4y – 3alternate method Vertex: (-1, -1) Opens to the left (narrow) Axis of symmetry: y = -1 x – intercept: (-3, 0) 9.1 Math 120 - KM

  20. The Distance Formula y c b x a 9.1 Math 120 - KM

  21. 9.1 Distance Formula Examples Determine the distance from P1 to P2. P1 (-2, 3) P2(2, 0) P1 (5, -2) P2(-3, -1) 9.1 Math 120 - KM

  22. 9.1 MIDPOINT Find the point exactly halfway from (-3,0) to (0,4) (0, 4) M(-1.5, 2) (0, 0) (-3, 0) 9.1 Math 120 - KM

  23. y x 9.1 Average the Coordinates! AVERAGE ! 9.1 Math 120 - KM

  24. 9.1 Midpoint Examples Determine the midpoint ofP1P2. P1 (-2, 3) P2(2, 0) P1 (5, -2) P2(-3, -1) 9.1 Math 120 - KM

  25. 9.1 Circles With a COMPASS How do I make a circle? 9.1 krm 11.2

  26. 9.1 Circle: Center (h,k) Radius r (h,k) r 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 krm 11.2

  27. 9.1 x2 + y2 = 1 Center: (0, 0) Radius: 1 The Unit Circle 9.1 Math 120 - KM

  28. 9.1 (x + 2)2 + (y – 4)2 = 32 Center: (-2, 4) Radius: 3 9.1 Math 120 - KM

  29. 9.1 x2 + (y + 4)2 = 25 Center: (0, - 4) Radius: 5 9.1 Math 120 - KM

  30. 9.1 Write the equation of the circle withradius 7 and center (-5, 8). 9.1 krm 11.2

  31. The Equation of a Circle How do I know it’s a circle? Look for ax2+ay2 9.1 krm 11.2

  32. Circle: Standard Form Write the equation of the circle in standard form and sketch the graph: x2 + y2 - 6x + 10y + 25 = 0 Center: (3, - 5) Radius: 3 9.1 krm 11.2

  33. 9.2 The Ellipse The Ellipse 9.2 Math 120 - KM

  34. 9.2 Ellipse (it fits in a box!) x-intercepts (+ a, 0) y-intercepts (0, + b) 9.2 Math 120 - KM

  35. 9.2 Example: Horizontal Major Axis 9.2 Math 120 - KM

  36. 9.2 Example: Vertical Major Axis 9.2 Math 120 - KM

  37. 9.2 Example: center not at the origin 9.2 Math 120 - KM

  38. 9.2 Example: Put in Standard Form First 9.2 Math 120 - KM

  39. 9.2 Example continued:Put in Standard Form First 9.2 Math 120 - KM

  40. 9.3 The Hyperbolait fits outside the box The Hyperbola X y 9.3 Math 120 - KM

  41. 9.3 The HyperbolaSTANDARD FORM X y 9.3 Math 120 - KM

  42. 9.3 Hyperbola: x2 is first • Fundamental Rectangle • Asymptotes • Vertices (if x2 – y2…) • Sketch X 9.3 Math 120 - KM

  43. 9.3 Example x2 is first 9.3 Math 120 - KM

  44. 9.3 Hyperbola: y2 is first • Fundamental Rectangle • Asymptotes • Vertices (if y2 – x2…) • Sketch y 9.3 Math 120 - KM

  45. 9.3 Example y2 is first y 9.3 Math 120 - KM

  46. 9.3 The HyperbolaNONSTANDARD FORM + + - - 9.3 Math 120 - KM

  47. 9.3 The HyperbolaNONSTANDARD FORMExample 1 9.3 Math 120 - KM

  48. 9.3 The HyperbolaNONSTANDARD FORMExample 2 9.3 Math 120 - KM

  49. Conics...2300+ years old? Math 120 - KM

  50. 9.4 Nonlinear Systems 9.4 Math 120 - KM

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