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Name______________________ Hr____ParabolasGeneral FormFactored FormVertex FormNew Definition:419100027114500Parts:VertexAxis of SymmetryFocusDirectrix5048250231775a=14p00a=14pFocal Diameter 2 Orientations:Up/DownLeft/RightExamples: Describe the graph y-12=4(x-1).Orientation:Vertex:Focus:Directrix:Axis of Symmetry:Sketch:22098027495500Write the equation for the parabola.Focus:Directrix:Write the equation for the parabola.40767016383000Focus:Directrix:Find the equation with Focus(-3, 4), directrix y=2 and graph it.10172702540000Graph the parabola: y2+4y=-x+1105219514033500Focus:Directrix:Find the equation with Vertex (-2, 3) and Focus (0, 3).12680956858000Directrix:Axis of symmetry:Graph the parabola: x2+6x-4y+1=0. 9759954445000Focus:Directrix:Engineers often design roads with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than on the sides. Find the equation of the parabola. (Place the vertex at the origin.)How far from the center of the road is the road surface 0.1 foot lower than in the middle?CirclesStandard FormGraphingEllipsesDefinitionStandard Formright325120PICTURE: Label the parts: Center Major Axis Minor Axis Vertices Co-Vertices Foci Equation Relating Different Radii:00PICTURE: Label the parts: Center Major Axis Minor Axis Vertices Co-Vertices Foci Equation Relating Different Radii:6096001016000398145021971000EXAMPLES: Find the equation of the ellipse.317515430500-6032521590000EXAMPLES: Graph the ellipse.x-224+y+328=12127254572000Minor Radius:Major Radius:Location of Foci:x+129+y-225=139687516891000Minor Radius:Major Radius:Location of Foci:Find the equation of an ellipse with Center (2, -3), one focus (3, -3) and one vertex (5, -3).86614019367500Find the equation of an ellipse with Vertices (±4, 0) and Foci (±2, 0).7854954699000Graph the ellipse: 4x2+y2-8x+4y+4=067564012763500Graph the ellipse: 3x2+5y2-12x+30y+42=0633095114363500Where is this applied? Newton’s work showed that objects in closed orbits must have circular or elliptical paths. However, if the velocity of an orbiting body is increased, its orbital path changes to a parabola or hyperbola, and it escapes the gravitational pull of the Sun and leaves the solar system.APPLICATION: Halley ’s Comet passed by Earth in 1986. It has an elliptical orbit with the sun at one focus. The eccentricity of the orbit is 0.97. The length of the major axis of the orbit is approximately 35.67 astronomical units. (An astronomical unit is about 93 million miles.) Find the standard form of the equation of the orbit. Place the center of the orbit at the origin and place the major axis on the x-axis. (NOTE: You will have a chance to view Halley’s Comet in 2061 when it passes by Earth again.) Diagram:Equation:How to find Eccentricity:Let’s use the equation for the Earth’s orbit from the journal article to compare the orbits of earth and Halley’s Comet (we’ll need to do some converting!)HyperbolasDefinitionStandard Formright288290PICTURE: Label the parts: Center Foci Transverse Axis Minor Axis Vertices/Co-vertices AsymptotesEquation Relating Two Radii:00PICTURE: Label the parts: Center Foci Transverse Axis Minor Axis Vertices/Co-vertices AsymptotesEquation Relating Two Radii:38690552317750022860021971000 EXAMPLES: Find the equation of the hyperbola.-19056921500Graph the hyperbola:x+229-y-5249=1Center:Vertices:Slope of asymptotes:139509516827500Foci:Determine the equation of the hyperbola having center Center (-1, 1), a vertex on the y-axis, and a focus at (-3, 1).1420495140335000Vertices:Slope of asymptotes:Graph the hyperbola: x2-4x-4y2-8y=1 Center:Vertices:Co-vertices:Foci:Slope of asymptotes:138874510541000Graph the hyperbola: 4x2-3y2+8x+16=0 Center:Vertices:Co-vertices:Foci:Slope of asymptotes:14973306223000Find the equation for the hyperbola with Foci (0, ±3), transverse axis length 4.48367958572500CLASSIFYING CONICS: Ax2+Cy2+Dx+Ey+F=0center0004876800229871PARABOLA0PARABOLA3686175315595206692577470Is A or C equal to 0?00Is A or C equal to 0?1438275315595523875134620START00START379984012700YES00YES2800350280670NO00NO28003502806704829175236220CIRCLECIRCLE213360083820Is A=C?00Is A=C?3867150147955YES0YES3686175857252800350239395NO00NO2800350220345202882533020Do A and C have the same sign?0Do A and C have the same sign?4876800125730ELLIPSE0ELLIPSE368617519621538290507620YES0YES2800350227965291465014605NO00NO237172550165HYPERBOLA0HYPERBOLA ................
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