Exploring Conic Sections: Parabolas & Nature's Paths

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. Intro to Conics • Conics Introduction Video • https://www.youtube.com/watch?v=HO2zAU3Eppo • How are conic sections formed? • What do we call the intersection of the two lines? • What is the name of the fixed vertical line? • What is the name of the rotating line? • What is a nappe? • What are the three conic sections formed? • Visual in xy • https://www.youtube.com/watch?v=GDHNoQHQmtQ • What do each degenerate to?

4. A Right Circular Cone (of two nappes)

5. Conic Sections andDegenerate Conic Sections

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

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

8. Parabola: Parabola Animation; https://www.youtube.com/watch?v=Im1qKj4nsqQWhen will I use this and explanation; https://www.youtube.com/watch?v=RwiflAmP6sU 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.

9. Graphs of x2 = 4py

10. Graphs of y2 = 4px

11. 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|

18. Example: Finding an Equation of a Parabola

19. Solution

20. 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|

21. Example: Finding an Equation of a Parabola

22. Solution