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4.4 Conics Recognize the equations and graph the four basic conics: parabolas, circles, ellipse, and hyperbolas. Write the equation and find the focus of a parabola. Write the equation of a ellipse and find the foci, vertices, the length of the major and minor axis.

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Parabolas Definition: A parabola is the set of all points (x, y) in a plane that are equidistant from a fixed line, called the directrix, and a fixed point, the focus, not on the line. Directrix

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Parabolas x 2 = 4py p 0 Vertex (0, 0) Directrix y = -p Focus (0, p) Line of sym x = 0 y 2 = 4px p 0 Vertex (0,0) Directrix x = -p Focus (p, 0) Line of sym y = 0 Standard equation of the Parabola

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Parabola Examples Given Find the focus.Since the squared variable is x, the parabola is oriented in the y directions. The leading coefficient is negative, so, the parabola opens down. -p p Focus (0, p) Solve for x 2 Solve for p. Focus

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Parabola Examples Write the standard form of the equation of the parabola with the vertex at the origin and the focus (2, 0). Note that the focus is along the x axis, so the parabola is oriented in the x axis direction, y 2 = 4px. Focus (2, 0) (p, 0) p = 2 y 2 = 4px = 4(2)x y 2 = 8x

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Classwork Page 370 problems 9 –14. Page 371 problems 17 – 28.

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Ellipse Definition: An ellipse is the set of all points (x, y ) in a plane the sum of whose distances from two distinct fixed points (foci) is constant. Focus (x, y) d1d1 d2d2 d 1 + d 2 = constant Vertex Major Axis Minor Axis Center

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Ellipse The standard form of the equation of an ellipse (Center at origin) where 0 < b < a c 2 = a 2 – b 2 Major axis length = 2a Minor axis length = 2b Major axis along the x axis Major axis along the y axis

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Ellipse Examples Given 4x 2 + y 2 = 36. Find the vertices, the end points of the minor axis, the foci and center. Change the equation to the standard form. a2a2 b2b2 Major axis along the y-axis Center (0, 0) Vertices (0, 6) End points of minor axis ( 3, 0) To find the foci use c 2 = a 2 – b 2 c 2 = 36 – 9 c 2 = 27 (0, a) ( b, 0) (0, c)

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Classwork Page 372 problems 35 – – 55

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Hyperbolas Definition: a hyperbola is the set of all points (x, y) the difference of whose distances from two distinct points (foci) is constant. (x, y) d1d1 d2d2 d 1 - d 2 = constant Center Vertex Focus Transversal Axis Branch

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