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10.1 Conics: Parabolas

10.1 Conics: Parabolas. Conic Sections. What is a conic section? A curve obtained as the intersection of a cone with a plane Traditionally, the 3 types of conic section are the parabola, ellipse, and hyperbola (the circle is a special case of the ellipse)

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10.1 Conics: Parabolas

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  1. 10.1 Conics: Parabolas

  2. Conic Sections • What is a conic section? • A curve obtained as the intersection of a cone with a plane • Traditionally, the 3 types of conic section are the parabola, ellipse, and hyperbola (the circle is a special case of the ellipse) • http://illuminations.nctm.org/ActivityDetail.aspx?ID=195

  3. Conic Sections • Where do these show up? • Whispering gallery • Microphones • Amphitheaters, band shells • Car headlights • Arches and bridges • Orbit of planets • Many others!

  4. Definition of a Parabola: A parabola is the set of all points (x,y) in a plane that are equidistant from a fixed line, the directrix, and a fixed point not on the line, the focus. The midpoint between the focus and directrix is the vertex. The line passing through the focus and the vertex is the axis of the parabola.

  5. The standard form of the equation of a parabola with vertex at (0,0) is as follows: vertical axis: +p opens up –p opens down horizontal axis: +p opens right –p opens left Play with the gizmo!!

  6. focus p -p directrix Parabola with vertex at (0,0) -Focus is at (0,p) -Directrix is at y = –p

  7. Ex 1) Find the vertex, focus, and directrix of the parabola and sketch its graph. rewrite equation: p = opens which direction? vertex: focus: directrix:

  8. Ex 2) Find the vertex, focus, and directrix of the parabola and sketch its graph. rewrite equation: p = opens which direction? vertex: focus: directrix:

  9. Ex 3) Find the vertex, focus, and directrix of the parabola and sketch its graph. rewrite equation: p = opens which direction? vertex: focus: directrix:

  10. Ex 4) Find the standard form of the equation of the parabola with vertex at the origin. opens which direction? gen equation: (3, -¾): p= eq. of par.:

  11. Ex 5) Find the standard form of the equation of the parabola with vertex at the origin. focus: opens which direction? gen equation: p= eq. of par.:

  12. Ex 6) Find the standard form of the equation of the parabola with vertex at the origin. focus: opens which direction? gen equation: p= eq. of par.:

  13. Ex 7) Find the standard form of the equation of the parabola with vertex at the origin. directrix: opens which direction? gen equation: p= eq. of par.:

  14. Ex 8) Find the standard form of the equation of the parabola with vertex at the origin. directrix: opens which direction? gen equation: p= eq. of par.:

  15. Ex 9) A satellite dish is shaped like a paraboloid of revolution. The signals that emanate from the satellite strike the surface of the dish and are reflected to single point, where the receiver is located. If the dish is 20 feet across at its opening and 6 feet deep at its center, at what position should the receiver be placed? gen. eq. w/vertex (0,0):

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