Demana Waits Foley Kennedy 8 1 Conic Sections

Demana, Waits, Foley, Kennedy 8. 1 Conic Sections and a New Look at Parabolas Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 1

What you’ll learn about n n 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. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 2

Intro to Conics n Conics Introduction Video n https: //www. youtube. com/watch? v=HO 2 z. AU 3 Eppo n 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 three conic sections formed? n Visual in xy n https: //www. youtube. com/watch? v=GDHNo. QHQmt. Q n What do each degenerate to? n n n Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 3

A Right Circular Cone (of two nappes) Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 4

Conic Sections and Degenerate Conic Sections Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 5

Conic Sections and Degenerate Conic Sections (cont’d) Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 6

Second-Degree (Quadratic) Equations in Two Variables Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 7

Parabola: Parabola Animation; https: //www. youtube. com/watch? v=Im 1 q. Kj 4 nsq. Q When will I use this and explanation; https: //www. youtube. com/watch? v=Rwifl. Am. P 6 s. U 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. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 8

Graphs of x 2 = 4 py Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 9

Graphs of y 2 = 4 px Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 10

Parabolas with Vertex (0, 0) n n n n Standard equation x 2 = 4 py Opens Upward or Focus Directrix Axis Focal length Focal width y 2 = 4 px downward To the right or to the left (0, p) y = –p y-axis p |4 p| (p, 0) x = –p x-axis p |4 p| Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 11

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 12

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 13

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 14

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 15

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 16

Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 17

Example: Finding an Equation of a Parabola Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 18

Solution Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 19

Parabolas with Vertex (h, k) n Standard equation (x– h)2 = 4 p(y – k)2 = 4 p(x – h) n Opens Upward or downward To the right or to the left Focus Directrix Axis Focal length Focal width (h, k + p) y = k-p x=h p |4 p| (h + p, k) x = h-p y=k p |4 p| n n n Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 20

Example: Finding an Equation of a Parabola Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 21

Solution Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 22