1. Conic Sections and a New Look at Parabolas
Demana, Waits, Foley, Kennedy
8.1
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 How are conic sections formed?
Conics Introduction Video
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
What do each degenerate to?

4. A Right Circular Cone (of two nappes)

5. Conic Sections and Degenerate Conic Sections

6. Conic Sections and Degenerate Conic Sections (cont’d)

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

8. Parabola: Parabola Animation; https://www. youtube. com/watch
Parabola: Parabola Animation; When will I use this and explanation;
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 = 4py y2 = 4px
Opens Upward or To the right or to the downward left
Focus (0, p) (p, 0)
Directrix y = –p x = –p
Axis y-axis x-axis
Focal length p p
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-p x = h-p
Axis x = h y = k
Focal length p p
Focal width |4p| |4p|

21. Example: Finding an Equation of a Parabola

22. Solution