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Exploring Parabolas: Conic Sections & Geometric Properties

Discover the beauty and significance of conic sections, focusing on the geometry and properties of parabolas. Learn about translations, reflective properties, and the nature of conic sections in relation to gravitational fields. Delve into the equations, vertex forms, and characteristics of parabolas, including standard forms and focal properties.

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Exploring Parabolas: Conic Sections & Geometric Properties

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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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