2 9.1 Intro to Conics EQUATION: EQUATIONS: A Circle is the set of all points, in a plane, such that each point is equidistant from a given point, the center.EQUATION:A Parabola is the set of all points, in a plane, that are equidistant from a given point, the focus, and a given line, the directrix.EQUATIONS:
3 9.1 cont’d. PICTURES Opens up/down Opens left/right Vertex : (h,k) Axis of Symmetry : x = h y = kFocus: (h, y+p) (h+p, k)Directrix: y = k – p x = h – pp is the distance from the vertex to the focus or vertex to the directrix. (p is directional)PICTURES
4 9.1 Examples Find the Standard Form Equation for each Parabola: 1. V (2, 3) Focus (2, 5) 2. V (-1, 4) D: x = 1Find Everything:Equation:Vertex:Focus:Directrix:
5 9.2 EllipsesAn Ellipse is the set of all points, in a plane, such that the sum of each points’ distances from two fixed points, the foci, is constant.Equations:Center : (h, k)Horizontal -Major Axis- VerticalVertical -Minor Axis- HorizontalC is the distance from the Center to the Foci.Length of Major Axis – 2a Length of Minor Axis – 2b
6 9.3 Hyperbolas Equations: A Hyperbola is the set of all points, in a plane, such that the difference of each points’ two distances from two fixed points, the foci, is constant.Equations:Center: (h,k)Right/Left OPENS Up/Downa is the distance from Center to vertices.“box” method for graphing, asymptotes.2 “parabolas”Slopes of asymptotes:
7 9.5 Parametric Equations Equations: If f and g are continuous functions of t on an interval, I.The set of ordered pairs (f(t), g(t)) is a plane curve, C.Parametric Equationst is the parameterEquations:Plot points in order of increasing values of t, you trace the curve in a specific direction. Orientation of the curve.Examples:Eliminating the Parameter1.Solve for t in one eq.2.Substitute into 2nd eq.3.Rectangular Equation
9 9.6/9.7 Polar Coordinates & Polar Graphs Directed distanceDirected anglerPlot points and give three equivalent points.Polar AxisCoordinate ConversionsRect. To PolarryxPolar to Rect,P to RR to PConvert:
10 9.6/9.7 cont’d. GRAPHS: Rose Curves Circle Line n is odd, n petals n is even, 2n petalsa is the length of petalsCirclesEquation ConversionNeed to know when sin and cos are (1 or -1).For sin, # petals = # of values.For cos, # petals need 1 more value.