Conic Sections and a New Look at Parabolas Demana, Waits, Foley, Kennedy 8.1 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 object in a gravitational field follows the path of a conic section.

Intro to Conics How are conic sections formed? 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?

A Right Circular Cone (of two nappes)

Conic Sections and Degenerate Conic Sections

Conic Sections and Degenerate Conic Sections (cont’d)

Second-Degree (Quadratic) Equations in Two Variables

Parabola: Parabola Animation; https://www. youtube. com/watch Parabola: Parabola Animation; https://www.youtube.com/watch?v=Im1qKj4nsqQ When 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.

Graphs of x2 = 4py

Graphs of y2 = 4px

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|

Example: Finding an Equation of a Parabola

Solution

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|

Example: Finding an Equation of a Parabola

Solution