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8.1. Demana, Waits, Foley, Kennedy. Conic Sections and a New Look at Parabolas. What you ’ ll learn about. Conic Sections Geometry of a Parabola Translations of Parabolas Reflective Property of a Parabola … and why Conic sections are the paths of nature: Any free-moving

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8.1

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  1. 8.1 Demana, Waits, Foley, Kennedy Conic Sections and a New Look at Parabolas

  2. What you’ll learn about • Conic Sections • Geometry of a Parabola • Translations of Parabolas • Reflective Property of a Parabola … and why Conic sections are the paths of nature: Any free-moving object in a gravitational field follows the path of a conic section.

  3. A Right Circular Cone (of two nappes)

  4. Conic Sections andDegenerate Conic Sections

  5. Conic Sections andDegenerate Conic Sections (cont’d)

  6. Second-Degree (Quadratic) Equations in Two Variables

  7. Parabola A parabola is the set of all points in a plane equidistant from a particular line (the directrix) and a particular point (the focus) in the plane.

  8. Graphs of x2 = 4py

  9. Parabolas with Vertex (0,0) • Standard equation x2 = 4pyy2 = 4px • Opens Upward or To the right or to the downward left • Focus (0, p) (p, 0) • Directrix y = –px = –p • Axis y-axis x-axis • Focal length pp • Focal width |4p| |4p|

  10. Graphs of y2 = 4px

  11. Example: Finding an Equation of a Parabola

  12. Solution

  13. Parabolas with Vertex (h,k) • Standard equation (x– h)2 = 4p(y – k)(y – k)2 = 4p(x – h) • Opens Upward or To the right or to the left downward • Focus (h, k + p) (h + p, k) • Directrix y = k-px = h-p • Axis x = h y = k • Focal length pp • Focal width |4p| |4p|

  14. Example: Finding an Equation of a Parabola

  15. Solution

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