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Mathematics 116 Bittinger Chapter 7 Conics

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Mathematics 116 Conics A conic is the intersection of a plane an a double- napped cone.

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Degenerate Conic Degenerate conic – plane passes through the vertex Point Line Two intersecting lines

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Algebraic Definition of Conic

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Definition of Conic Locus (collection) of points satisfying a certain geometric property.

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Circle A circle is the set of all points (x,y) that are equidistant from a fixed point (h,k) The fixed point is the center. The fixed distance is the radius

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Algebraic def of Circle Center is (h,k) Radius is r

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Equation of circle with center at origin

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Def: Parabola A parabola is the set of all points (x,y) that are equidistant from a fixed line, the directrix, and a fixed point, the focus, not on the line.

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Standard Equation of Parabola Vertex at Origin Vertex at (0,0) Directrix y = -p Vertical axis of symmetry

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Standard Equation of Parabola Opening left and right Vertex: (0,0O Directrix: x = -p Axis of symmetry is horizontal

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Willa Cather – U.S. novelist (1873-1947) “The higher processes are all simplification.”

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Definition: Ellipse An ellipse is the set of all points (x,y), the sum of whose distances from two distinct points (foci) is a constant.

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Standard Equation of Ellipse Center at Origin Major or focal axis is horizontal

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Standard Equation of Ellipse Center at Origin Focal axis is vertical

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Ellipse: Finding a or b or c

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Definition: hyperbola A hyperbola is the set of all points (x,y) in a plane, the difference whose distances from two distinct fixed points (foci) is a positive constant.

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Hyperbola equation opening left and right centered at origin

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Standard Equation of Hyperbola opening up and down centered at origin

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Hyperbola finding a or b or c

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Objective – Conics centered at origin Recognize, graph and write equations of Circle Parabola Ellipse Hyperbola –Find focal points

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Rose Hoffman – elementary schoolteacher “Discipline is the keynote to learning. Discipline has been the great factor in my life.”

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Mathematics 116 Translations Of Conics

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Circle Center at (h,k)radius = r

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Ellipse major axis horizontal

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Ellipse major axis vertical

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Hyperbola opening left and right

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Hyperbola opening up and down

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Parabola vertex (h,k) opening up and down

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Parabola vertex (h,k) opening left and right

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Objective Recognize equations of conics that have been shifted vertically and/or horizontally in the plane.

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Objective Find the standard form of a conic – circle, parabola, ellipse, or hyperbola given general algebraic equation.

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Example Determine standard form – sketch Find domain, range, focal points

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Example - problem Determine standard form – sketch Find domain, range, focal points

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Winston Churchill “It’s not enough that we do our best; sometimes we have to do what’s required.”

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