conic sections project by: andrew pistana 1 st hour honors algebra 2

Conic Sections Project

By: Andrew Pistana

1st Hour

Honors Algebra 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.

Conic SectionsClick on this site for a fun, interactive applet!!http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/awl/conics-main.html

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

Different Forms Of Conic Sections

• Click on one of these buttons to learn more about that form of Conic Section.

Parabolas

HyperbolasCircles

Ellipses

THEEND

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)

Parabolas

Parabola LinksClick here to go back to different forms of Conic Sections!

•http://en.wikipedia.org/wiki/Derivations_of_conic_sections

•http://etc.usf.edu/clipart/galleries/math/conic_parabolas.php

•http://analyzemath.com/parabola/FindEqParabola.html

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

Ellipses

Ellipses•Video: http://www.bing.com/videos/search?q=conic+sections+ellipse&view=detail&mid=05D6AFE5CF2D689E455F05D6AFE5CF2D689E455F&first=0&FORM=LKVR19

Ellipses

• Useful Links:

• http://mathforum.org/library/drmath/view/62576.html

• http://en.wikipedia.org/wiki/Ellipse

• http://mathworld.wolfram.com/Ellipse.html Back to different forms of Conic Sections

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)

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

Circles

Circles

• http://www.mathwarehouse.com/geometry/circle/equation-of-a-circle.php

• http://en.wikipedia.org/wiki/Circle

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.

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:

Hyperbolas

• http://www.youtube.com/watch?v=Z6cwpsDC_5A

Hyperbolas

• http://www.analyzemath.com/EquationHyperbola/EquationHyperbola.html

• http://www.slu.edu/classes/maymk/GeoGebra/EllipseHyperbola.html

• http://en.wikipedia.org/wiki/Hyperbola

THE END

• Thank You for looking through my presentation!