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- 1. Conic Sections Project By: Andrew Pistana 1st Hour Honors Algebra 2
- 2. Conic Sections • A conic section is a geometric curve formed by cutting a cone. A curve produced by the intersection of a plane with a circular cone. Some examples of conic sections are parabolas, ellipses, circles, and hyperbolas.
- 3. Conic Sections Click on this site for a fun, interactive applet!! http://cs.jsu.edu/mcis/faculty/leath rum/Mathlets/awl/conics- main.html
- 4. Conic Sections • Learn more about Conic Sections on these websites! • http://en.wikipedia.org/wiki/Conic_section • http://math2.org/math/algebra/conics.htm • http://xahlee.org/SpecialPlaneCurves_dir/ ConicSections_dir/conicSections.html
- 5. Different Forms Of Conic Sections • Click on one of these buttons to learn more about that form of Conic Section. Parabolas Hyperbolas Circles Ellipses THE END
- 6. Parabolas • A parabola is a mathematical curve, formed by the intersection of a cone with a plane parallel to its side. Equation Focus Directrix Axis of Symmetry x2 = 4py (0,p) y = -p Vertical (x = 0) y2 = 4px (p,0) x = -p Horizontal (y = 0)
- 7. Parabolas
- 8. Parabola Links Click here to go back to different forms of Conic Sections! •http://en.wikipedia.org/wiki/Derivati ons_of_conic_sections •http://etc.usf.edu/clipart/galleries/m ath/conic_parabolas.php •http://analyzemath.com/parabola/Fi ndEqParabola.html
- 9. Ellipses • An ellipse is an intersection of a cone and oblique plane that does not intersect the base of the cone. • Standard Form Vertices: (+/-a,0) (0,+/-a) Co-Vertices: (0,+/-b) (+/-b,0) When finding the foci, use the following equation…. c2 = a2 – b2
- 10. Ellipses
- 12. Ellipses • Useful Links: • http://mathforum.org/library/drmath/view/6 2576.html • http://en.wikipedia.org/wiki/Ellipse • http://mathworld.wolfram.com/Ellipse.html Back to different forms of Conic Sections
- 13. Circles • Definition: A circle is the set of all points that are the same distance, r, from a fixed point. General Formula: X2 + Y2=r2 where r is the radius • Unlike parabolas, circles ALWAYS have X2 and Y 2 terms. – X2 + Y2=4 is a circle with a radius of 2 ( since 4 =22)
- 14. Circle Example Problem • What is the equation of the circle pictured on the graph below? Answer Since the radius of this this circle is 1, and its center is the origin, this picture's equation is (Y-0)² +(X-0)² = 1 ² Y² + X² = 1
- 15. Circles
- 17. Hyperbolas • A hyperbola is a conic section formed by a point that moves in a plane so that the difference in its distance from two fixed points in the plane remains constant.
- 18. Hyperbolas • Focus of hyperbola : the two points on the transverse axis. These points are what controls the entire shape of the hyperbola since the hyperbola's graph is made up of all points, P, such that the distance between P and the two foci are equal. To determine the foci you can use the formula: a2 + b2 = c2 • Transverse axis: this is the axis on which the two foci are. • Asymptotes: the two lines that the hyperbolas come closer and closer to touching. The asymptotes are colored red in the graphs below and the equation of the asymptotes is always:
- 20. Hyperbolas • http://www.analyzemath.com/EquationHyp erbola/EquationHyperbola.html • http://www.slu.edu/classes/maymk/GeoGe bra/EllipseHyperbola.html • http://en.wikipedia.org/wiki/Hyperbola
- 21. THE END • Thank You for looking through my presentation!