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Notes 8.1 Conics Sections – The Parabola

Notes 8.1 Conics Sections – The Parabola. I. Introduction. A.) A conic section is the intersection of a plane and a cone . B.) By changing the angle and the location of intersection, a parabola, ellipse, hyperbola, circle, point, line, or a pair of intersecting lines is produced.

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Notes 8.1 Conics Sections – The Parabola

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  1. Notes 8.1 Conics Sections – The Parabola

  2. I. Introduction A.) A conic section is the intersection of a plane and a cone. B.) By changing the angle and the location of intersection, a parabola, ellipse, hyperbola, circle, point, line, or a pair of intersecting lines is produced.

  3. C.) Standard Conics: 1.) Parabola 2.) Ellipse 3.) Hyperbola

  4. D.) Degenerate Conics 1.) Circle 2.) Point 3.) Line 4.) Intersecting Lines

  5. E.) Forming a Parabola – When a plane intersects a double-napped cone and is parallel to the side of the cone, a parabola is formed.

  6. F.) General Form Equation for All Conics If both B and C = 0, or A and B = 0, the conic is a parabola

  7. II. The Parabola A.) In general - A parabola is the graph of a quadratic equation, or any equation in the form of

  8. B.) Def. - 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.

  9. Axis of Symmetry Focus Focal Width Vertex Focal Length Directrix

  10. C.) Parabolas (Vertex = (0,0)) Standard Form Focus Directrix Axis of Symmetry Focal Length Focal Width

  11. D.) Ex. 1- Find the focus, directrix, and focal width of the parabola y = 2x2. Focus = Directrix = Focal Width =

  12. E.) Ex. 2- Do the same for the parabola Focus = Directrix = Focal Width =

  13. F.) Ex. 3- Find the equation of a parabola with a directrix of x = -3 and a focus of (3, 0).

  14. G.) Parabolas (Vertex = (h, k)) St. Fm. Focus Directrix Ax. of Sym. Fo. Lgth. Fo. Wth.

  15. H.) Ex. 4- Find the standard form equation for the parabola with a vertex of (4, 7) and a focus of (4, 3).

  16. I.) Ex. 5- Find the vertex, focus, and directrix of the parabola 0 = x2 – 2x – 3y – 5. focus = Directrix = vertex =

  17. III. Paraboloids of Revolution A.) A PARABOLOID is a 3-dimensional solids created by revolving a parabola about an axis. Examples of these include headlights, flashlights, microphones, and satellites.

  18. B.) Ex. 6– A searchlight is in the shape of a paraboloid of revolution. If the light is 2 feet across and 1 ½ feet deep, where should the bulb be placed to maximize the amount of light emitted?

  19. The bulb should be placed 2” from the vertex of the paraboloid

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