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. ”