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Introduction to Conics: Understanding Parabolas Algebraically

Learn about parabolas, graphs of quadratic equations in two variables, using algebraic techniques. Explore vertex, axis, and equations of parabolas to master conics. Impress with examples and reflective properties.

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Introduction to Conics: Understanding Parabolas Algebraically

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  1. Notes #3-___Date:______ 8.1 Introduction to Conics: Parabolas (632) Defined algebraically as the graphs of second degree (quadratic) equations in two variables: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, where A, B & C are not all 0.* Completing the square is used to get the standard equation. Generator Vertex Nappes (the two cones) Axis

  2. Page 633, Figure 8.2 Conic Sections (conics): Degenerate Conics: A circle is considered to be degenerate ellipse.

  3. Equations of Parabolas • General Form: y = ax2 + 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:

  4. Definition of a Parabola: • Algebra in Motion, Conics: parabola 1

  5. Page 634, Figure 8.3 p p

  6. Page 637 – Don’t copy this down, it’s in your book!

  7. Example 1: Always sketch the given information! Find the vertex, focus & directrix of: a) x2 – 6x – 4y + 5 = 0 b) y2 + 6y + 8x + 25 = 0 V(3, -1), f(3, 0) & y = -2 V(-2, -3), f(-4, -3) & x = 0

  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) y2 = -4(x – 4) (y – 2)2 = 8x 5x2 = 9y

  9. Page 639 • Reflective Property of a Parabola

  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

  11. Assignment • A#3-20: (641) #1-62 D2S2, 65-70

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