Mathematics 116 Bittinger Chapter 7 Conics Mathematics 116
Mathematics 116 Bittinger • Chapter 7 • Conics
Mathematics 116 • Conics • A conic is the intersection of a plane an a doublenapped cone.
Degenerate Conic • Degenerate conic – plane passes through the vertex • Point • Line • Two intersecting lines
Algebraic Definition of Conic
Definition of Conic • Locus (collection) of points satisfying a certain geometric property.
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
Algebraic def of Circle • Center is (h, k) • Radius is r
Equation of circle with center at origin
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.
Standard Equation of Parabola Vertex at Origin • Vertex at (0, 0) • Directrix y = -p • Vertical axis of symmetry
Standard Equation of Parabola Opening left and right • Vertex: (0, 0 O • Directrix: x = -p • Axis of symmetry is horizontal
Willa Cather – U. S. novelist (1873 -1947) • “The higher processes are all simplification. ”
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.
Standard Equation of Ellipse Center at Origin • Major or focal axis is horizontal
Standard Equation of Ellipse Center at Origin • Focal axis is vertical
Ellipse: Finding a or b or c
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.
Hyperbola equation opening left and right centered at origin
Standard Equation of Hyperbola opening up and down centered at origin
Hyperbola finding a or b or c
Objective – Conics centered at origin • Recognize, graph and write equations of • • Circle Parabola Ellipse Hyperbola – Find focal points
Rose Hoffman – elementary schoolteacher • “Discipline is the keynote to learning. Discipline has been the great factor in my life. ”
Mathematics 116 • Translations • Of • Conics
Circle • Center at (h, k) radius = r
Ellipse major axis horizontal
Ellipse major axis vertical
Hyperbola opening left and right
Hyperbola opening up and down
Parabola vertex (h, k) opening up and down
Parabola vertex (h, k) opening left and right
Objective • Recognize equations of conics that have been shifted vertically and/or horizontally in the plane.
Objective • Find the standard form of a conic – circle, parabola, ellipse, or hyperbola given general algebraic equation.
Example • Determine standard form – sketch • Find domain, range, focal points
Example - problem • Determine standard form – sketch • Find domain, range, focal points
Winston Churchill • “It’s not enough that we do our best; sometimes we have to do what’s required. ”
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