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Mathematics 116 Bittinger Chapter 7 Conics
Mathematics 116 Conics A conic is the intersection of a plane an a double- napped 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: (0,0O 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.”
4.4 Conics Recognize the equations and graph the four basic conics: parabolas, circles, ellipse, and hyperbolas. Write the equation and find the focus.
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Section 11.6 – Conic Sections
What do we know about parabolas?. Conic Slice Algebraic Definition Parabola: For a given point, called the focus, and a given line not through the focus,
Conics, Parametric Equations, and Polar Coordinates 10 Copyright © Cengage Learning. All rights reserved.
Conic Sections MAT 182 Chapter 11
10.1 Conics and Calculus. Each conic section (or simply conic) can be described as the intersection of a plane and a double-napped cone. CircleParabolaEllipse.
Conic Sections Parabola Ellipse Hyperbola
Chapter 9 Analytic Geometry.
LEARNING TARGETS AFTER YOU COMPLETE THIS CHAPTER, YOU WILL BE ABLE TO:
Conic Sections Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Conic Sections Conic sections are plane figures formed.
Conics, Parametric Equations, and Polar Coordinates
Ch. 9 Objective: Understand and identify basic characteristics of conics. Conic section (conic): What you get (the intersection)when you cross a.
The Second Degree Equations. Determine whether the following equations represent a circle, parabola, ellipse or hyperbola. 1. x 2 + 4y 2 – 6x + 10y.
C.P. Algebra II The Conic Sections Index The Conics The Conics Translations Completing the Square Completing the Square Classifying Conics Classifying.
CHAPTER 9 CONIC SECTIONS.
Hyperbolas 9.3. Definition of a Hyperbola A hyperbola is the set of all points (x, y) in a plane, the difference of whose distances from two distinct.
9.1 Conic Sections Conic sections – curves that result from the intersection of a right circular cone and a plane. Circle Ellipse Parabola Hyperbola.
What is the standard form of a parabola who has a focus of ( 1,5) and a directrix of y=11.
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