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1 Copyright © 2004 Pearson Education, Inc. Notes #3-___ Notes #3-___ Date:______ Date:______ Vertex Nappes (the two cones) Generato r Axis 8.1 Introduction to Conics: 8.1 Introduction to Conics: Parabolas (632) Parabolas (632) Defined algebraically as the graphs of second degree Defined algebraically as the graphs of second degree (quadratic) equations in two variables: (quadratic) equations in two variables: Ax Ax 2 + Bxy + Cy + Bxy + Cy 2 + + Dx + Ey + F = 0, Dx + Ey + F = 0, where A, B & C are not all 0. where A, B & C are not all 0. * Completing the square is used to get the standard equation. * Completing the square is used to get the standard equation.
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Notes #3-___ Date:______. 8.1 Introduction to Conics: Parabolas (632). - PowerPoint PPT Presentation
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Notes #3-___Notes #3-___Date:______Date:______
VertexNappes (the two cones)
Generator
Axis
8.1 Introduction to Conics: Parabolas (632)8.1 Introduction to Conics: Parabolas (632)
Defined algebraically as the graphs of second degree (quadratic) equations in two Defined algebraically as the graphs of second degree (quadratic) equations in two
variables: variables: AxAx22 + Bxy + Cy + Bxy + Cy22 + Dx + Ey + F = 0, + Dx + Ey + F = 0, where A, B & C are not all 0.where A, B & C are not all 0.
* Completing the square is used to get the standard equation.* Completing the square is used to get the standard equation.
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Page 633, Figure 8.2Page 633, Figure 8.2
Degenerate Conics:
A circle is considered to
be degenerate ellipse.
Conic Sections (conics):
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Equations of ParabolasEquations of Parabolas
General Form: General Form: y = axy = ax22 + bx + c + bx + c
Vertex:Vertex:
y-intercept:y-intercept: Standard Form: y = a(x – h)Standard Form: y = a(x – h)22 + k + k
y – k = a(x – h)y – k = a(x – h)22
Vertex:Vertex:
The sign of “a” in both determines:The sign of “a” in both determines:
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Definition of a Parabola:Definition of a Parabola:
Algebra in Motion, Conics: parabola 1Algebra in Motion, Conics: parabola 1
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Page 634, Figure 8.3Page 634, Figure 8.3
14
ap
p
2
2
14
14
y a x h k
ap
y x h kp
p
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Page 637 – Don’t copy this down, it’s in your book!Page 637 – Don’t copy this down, it’s in your book!
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Example 1Example 1::Always sketch the given information!Always sketch the given information!
Find the vertex, focus & directrix of:Find the vertex, focus & directrix of:
a) xa) x22 – 6x – 4y + 5 = 0 – 6x – 4y + 5 = 0
b) yb) y22 + 6y + 8x + 25 = 0 + 6y + 8x + 25 = 0
V(3, -1), f(3, 0) & y = -2
V(-2, -3), f(-4, -3) & x = 0
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Example 2Example 2::Write an equation in standard form if:Write an equation in standard form if:
a) Vertex: (4, 0) & directrix: x = 5a) Vertex: (4, 0) & directrix: x = 5
b) Focus: (2, 2) & directrix: x = -2b) Focus: (2, 2) & directrix: x = -2
c)c) Vertex: (0, 0) & through (3, 5)Vertex: (0, 0) & through (3, 5)
y2 = -4(x – 4)
(y – 2)2 = 8x
5x2 = 9y
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Page 639Page 639
Reflective Property of a ParabolaReflective Property of a Parabola
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Example 3Example 3::The satellite dishes at the VLA are 82 The satellite dishes at the VLA are 82 feet in diameter. If they have a depth feet in diameter. If they have a depth of 10.25 feet, how far from the vertex of 10.25 feet, how far from the vertex should the receiving antenna be should the receiving antenna be placed?placed?
p = 41 feet
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AssignmentAssignment
A#3-20: (641) #1-62 D2S2, 65-70A#3-20: (641) #1-62 D2S2, 65-70
