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Objective: Understand and identify basic characteristics of conics. Conic section (conic): What you get (the intersection)when you cross a plane and a double-napped cone. 4 Basic Conics: Vertex Axis ELLIPSE: The plane is slightly tilted so it’s no longer perpendicular to the axis. CIRCLE: The plane is exactly perpendicular to the cone’s axis. PARABOLA: Keep tilting so that the plane is now exactly parallel to the side of the top cone. (Parabola occurs because one side of the ellipse sort of falls off.) HYPERBOLA: Keep tilting so that the plane is now slicing though both the top and bottom parts of the cone. CONICS (Pre/Calc Style): Conic: A __________________of points satisfying a certain geometric property. Ex: A circle is the locus of all points equidistant from a fixed center point. 9.1 Parabolas Parabola (Conical Definition): ________ ________ The set of all points (x, y) in a plane that are __________ from a fixed line, the _______ (parallel to the x or y-axis), and a fixed point (not on the line), called the _________. The midpoint between the focus and the directrix. The line passing through the focus and the vertex. (h, k) Standard form If the axis is _____________________ (x is squared): of the equation of a parabola with vertex at (h, k) p is the ____________________ (can be positive or negative) from the vertex to the focus Note: p≠0 V F F V P _______ P _____ Vertex: ___________ Focus: ___________ Axis of Symmetry: ________ Directrix: __________ Standard form If the axis is ________________ __________(y is squared) of the equation of a parabola with vertex at (h, k) p is the directed distance (can be positive or negative) from the vertex to the focus V F F V Note: p≠0 P _____ P _____ Vertex: ___________ Focus: ___________ Axis of Symmetry: _________ Directrix: _________ **Determine Characteristics and sketch graphs** Given the equation of a parabola, identify its a. b. c. d. Vertex Focus Axis of symmetry Directrix Hint: Determine orientation of the parabola and p first. Ex. 1) **Determine Characteristics and sketch graphs** Given the equation of a parabola, identify its a. b. c. d. Vertex Focus Axis of symmetry Directrix Hint: Determine orientation of the parabola and p first. Ex. 2) HW : For each parabolic equation, identify (and sketch) the parabola’s : a) Vertex b) Focus c) Axis of symmetry d) Directrix.