2. Copyright © 2004 Pearson Education, Inc. 1 Notes #3-___ Date:______ Vertex Nappes (the two cones) Generator Axis 8.1 Introduction to Conics: Parabolas (632) Defined algebraically as the graphs of second degree (quadratic) equations in two variables: Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, where A, B & C are not all 0. * Completing the square is used to get the standard equation.

3. Copyright © 2004 Pearson Education, Inc. 2 Page 633, Figure 8.2 Degenerate Conics: A circle is considered to be degenerate ellipse. Conic Sections (conics):

4. Copyright © 2004 Pearson Education, Inc. 3 Equations of Parabolas  General Form: y = ax 2 + bx + c Vertex:y-intercept:  Standard Form: y = a(x – h) 2 + k y – k = a(x – h) 2 Vertex: The sign of “a” in both determines:

5. Copyright © 2004 Pearson Education, Inc. 4 Definition of a Parabola:  Algebra in Motion, Conics: parabola 1 Algebra in Motion, Conics: parabola 1 Algebra in Motion, Conics: parabola 1

6. Copyright © 2004 Pearson Education, Inc. 5 Page 634, Figure 8.3 p p

7. Copyright © 2004 Pearson Education, Inc. 6 Page 637 – Don’t copy this down, it’s in your book!

8. Copyright © 2004 Pearson Education, Inc. 7 Example 1: Always sketch the given information! Find the vertex, focus & directrix of: a) x 2 – 6x – 4y + 5 = 0 b) y 2 + 6y + 8x + 25 = 0 V(3, -1), f(3, 0) & y = -2 V(-2, -3), f(-4, -3) & x = 0

9. Copyright © 2004 Pearson Education, Inc. 8 Example 2: Write an equation in standard form if: a) Vertex: (4, 0) & directrix: x = 5 b) Focus: (2, 2) & directrix: x = -2 c)Vertex: (0, 0) & through (3, 5) y 2 = -4(x – 4) (y – 2) 2 = 8x 5x 2 = 9y

10. Copyright © 2004 Pearson Education, Inc. 9 Page 639  Reflective Property of a Parabola Reflective Property of a Parabola Reflective Property of a Parabola

11. Copyright © 2004 Pearson Education, Inc. 10 Example 3: The satellite dishes at the VLA are 82 feet in diameter. If they have a depth of 10.25 feet, how far from the vertex should the receiving antenna be placed? p = 41 feet

12. Copyright © 2004 Pearson Education, Inc. 11 Assignment  A#3-20: (641) #1-62 D2S2, 65-70