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Math Project

Math Project. Parabola. Degenerate Case. Limiting case in which a class of objects changes its nature so as to belong to another simpler class ex. A point is a degenerate circle with radius one ex. A line is a degenerate form of parabola. Application.

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Math Project

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  1. Math Project Parabola

  2. Degenerate Case • Limiting case in which a class of objects changes its nature so as to belong to another simpler class • ex. A point is a degenerate circle with radius one • ex. A line is a degenerate form of parabola

  3. Application • Parabolas have a property that if they are made of a material that reflects light, the light enters the parabola traveling parallel to it axis of symmetry and is reflected to the focus. • Ex. Head light reflectors, telescopes, bridges, search lights, and satellite dishes.

  4. Standard form • Common equation • ex. Y=Ax²+Bx+C

  5. Geometry and Algebra II Form • Standard- Y=Ax²+Bx+C • Vertex- Y=a(x-h)²+k • Intercept- Y=a(x-h)(x-k)

  6. Key Words • Square function • Parabola • Intercept • Vertex • Axis of symmetry • Minimum/Maximum • Focus • Directrix • Focal distance

  7. Relationships • Vertex form- y=a(x-h)²-k • a›0 opens up/ a‹o opens down • h=horizontal shift • k=vertical shift • a=vertical stretch or compression

  8. Rotated form • y²=4ax • (y-k)²=4a(x-h)

  9. Conic form • 4a(y-k)=(x-h)² • x²=4ay

  10. Eccentricity • E= c/a • A=distance from center to vertex • C=distance from center to focus • Eccentricity=1 for a parabola

  11. Labeled Graph

  12. Construction & Origin • The parabola is defined as the locus of a point which moves so that it is always the same distance from a fixed point (called the focus) and a given line (called the directrix). • [The word locus means the set of points satisfying a given condition. See some background in Distance from a Point to a Line.] • In the following graph, • The focus of the parabola is at (0,p)  .

  13. Construction & Origin • The directrix is the line y=−p  . • The focal distance is |p|  (Distance from the origin to the focus, and from the origin to the directrix. We take absolute value because distance is positive.) • The point (x, y) represents any point on the curve. • The distanced from any point (x, y) to the focus (0,p)  is the same as the distance from (x, y) to the directrix.

  14. Construction & Origin Conic section: Parabola All of the graphs in this chapter are examples of conic sections. This means we can obtain each shape by slicing a cone at different angles. How can we obtain a parabola from slicing a cone? We start with a double cone (2 right circular cones placed apex to apex): If we slice a cone parallel to the slant edge of the cone, the resulting shape is a parabola, as shown.

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