PP Section 11.2 - Horizontal Parabolas; notice that these...

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Conic Sections
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Conic sections, or conics, get their name because they result from intersecting a cone with a plane.
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PARABOLA A parabola is the set of points in a plane that are equidistant from a fixed point F (the focus) and a fixed line (the directrix). F The line through the focus perpendicular to the directrix is called the axis of symmetry of the parabola. Directrix
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F V The midpoint between the focus and the directrix is called the vertex V of the parabola. In this case, the vertex is at the origin. D: Directrix
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Equation of a Parabola To obtain an equation for a parabola, we place its vertex at the origin O and its directrix parallel to the x -axis. We will use the definition of parabola to get its equation. Assume: F(0, p), P(x, y), D: y = -p V P F D
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The defining property of a parabola is that the distance to the focus equals the distance to the directrix. 2 2 ( ) x y p y p + - = +
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Hence, 2 2 2 2 2 2 2 2 2 2 ( ) ( ) 2 2 4 x y p y p y p x y py p y py p x py + - = + = + + - + = + + = 2 2 ( ) x y p y p + - = +
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Similar arguments yield the equations of the other three possibilities. Vertical Parabolas
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Unformatted text preview: Horizontal Parabolas; notice that these curves do NOT represent functions. Example Find the focus and directrix of the parabola y 2 + 10 x = 0 and sketch the graph. 2 5 2 5 2 ) ( 4 10-= -=-= p x x y 2 5 2 5 : ), , ( =-= x D F Horizontal parabola Example Find the focus, vertex, and directrix of the parabola ( y+1) 2 =-4(x 2) and sketch the graph. A horizontal and a vertical translation have taken place. The vertex is now (2, -1) and p = -1. The parabola opens left, so to find its focus we need to move 1 unit to the left of the vertex. Hence, F = (1, -1) The directrix is 1 unit to the right of the focus. Hence, D: x = 3. F V D Definitions A chord of a parabola is a straight line segment joining any two points on the curve. If a chord passes through the focus, it is called a focal chord. The focal width is the length of the focal chord perpendicular to the axis of symmetry. The focal width is always 4p ....
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PP Section 11.2 - Horizontal Parabolas; notice that these...

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