Copyright © 2000 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen College Algebra: A Graphing Approach Chapter Seven Additional Topics in Analytical Geometry

Copyright © 2000 by the McGraw-Hill Companies, Inc. Circle Ellipse Parabola Hyperbola Conic Sections

Copyright © 2000 by the McGraw-Hill Companies, Inc. 1. y 2 = 4ax Vertex: (0, 0) Focus: (a, 0) Directrix: x = –a Symmetric with respect to the x axis. Axis the x axis 2. x 2 = 4ay Vertex: (0, 0) Focus: (0, a) Directrix: y = –a Symmetric with respect to the y axis. Axis the y axis a 0 (opens right) a 0 (opens up) Standard Equations of a Parabola with Vertex at (0, 0)

Copyright © 2000 by the McGraw-Hill Companies, Inc. Standard Equations of an Ellipse with Center at (0, 0) [Note: Both graphs are symmetric with respect to the x axis, y axis, and origin. Also, the major axis is always longer than the minor axis.]

Copyright © 2000 by the McGraw-Hill Companies, Inc. 2. y 2 a 2 – x 2 b 2 = 1 x intercepts: none y intercepts: ± a (vertices) Foci: F' (0, – c ) F (0, c ) c 2 = a 2 + b 2 Transverse axis length = 2 a Conjugate axis length = 2 b 1. x 2 a 2 + y 2 b 2 = 1 x intercepts: ± a (vertices) y intercepts: none Foci: F' (– c, 0) F ( c c 2 = a 2 + b 2 Transverse axis length = 2 a Conjugate axis length = 2 b [Note: Both graphs are symmetric with respect to the x axis, y axis, and origin.] Standard Equations of a Hyperbola with Center at (0, 0)

Copyright © 2000 by the McGraw-Hill Companies, Inc. Circles (x – h) 2 + (y – k) 2 = r 2 Center (h, k) Radius r (x – h) 2 = 4a(y – k) Vertex (h, k) Focus (h, k + a) a > 0 opens up a < 0 opens down (y – k) 2 = 4a(x – h) Vertex (h, k) Focus (h + a, k) a < 0 opens left a > 0 opens right Parabolas Standard Equations for Translated Conics—I

Copyright © 2000 by the McGraw-Hill Companies, Inc. Ellipses Standard Equations for Translated Conics—II (a)

Copyright © 2000 by the McGraw-Hill Companies, Inc. Hyperbolas Standard Equations for Translated Conics—II (b)

Copyright © 2000 by the McGraw-Hill Companies, Inc. x = (v 0 cos  ) t y = a 0 + (v 0 sin  ) t – 4.9 t 2 Projectile Motion