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Bria Thorne Conic Sections
Transcript of Bria Thorne Conic Sections
2-26-13 Circle examples Def.
A round plan figure whose boundary consists of points equidistant from a fixed center. Circle The word itself comes from the Greek word, kirkos meaning "a circle". The circle originates from the beginning of recorded history. People have observed circles in everyday life from the beginning of life, for example, the Sun and Moon.
The circle was the base for all future math. It helped develop geometry, astronomy and calculus. Circle History A few examples of circles in real life would be the distance around a track field for all the sports people. An example for a car person would be the distance around a car tire, finding this would shows how far one roll is. Last but not least all of my people who need an example pertaining to food to get something think about the surface area of a pizza. Real life situations First and foremost you must know the equation of a circle it is something you should never forget in your life. (x-h)^2 + (y-k)^2=r^2 Ellipse Def.
An ellipse obtained as the intersection of a cone with an inclined plane.
Two of the four distances have to be the same. Ellipse History First the ellipse is a complex shape. It has 2 MAIN points, the focus and directrix. Kelper, from Greece, believed that the orbit of Mars was oval. Most Math comes from Greece or Africa. Anyway Kelper later discovered that Mars was in an ellipse with the Sun at one focus. Thus the focus of an ellipse was born.
Also if you haven't noticed by now most geometry comes from studying the galaxy. Ellipse in Real life Simple Ellipse: Race track designs that help designers take into account top speeds and such depending on shape. First one must know the equations for ellipses and their focis and directrixes. The foci and directrix are equidistant from the vertex but in opposite directions. There are also two ways in which an ellipse can be facing, giving you multiple functions for an ellipse.
The ellipse going horizontal on the plane fuction's would look like:
x^2\a^2 + y^2/b^2 Foci- (c,0), (-c,0)
Vertices- (a,0) (-a,0) Co-vertices- (0,b) (0,-b)
The ellipse going vertical on the plane fuction's would look like:
x^2\b^2 + y^2/a^2 Foci- (0,c), (0,-c)
Vertices- (0,a) (0,-a) Co-vertices- (b,0) (-b,0)
Seems complicated? It's not. Conic sections are really the easiest things we will do this year. (see next slide for pictures to help explain) Simple Ellipse cont. Still confused? All you are doing is imputing numbers into a function and doing simple division and/or addition. Please do not let the letters and numbers together confuse you. Please see next slide for examples. Ellipse examples:
a = 5
b = 3
a = 3
b = 2 Parabola Def.
A parabola is a curve where any point is at an equal distance from:
•a fixed point (the focus), and
•a fixed straight line (the directrix) Parabola history Menaechmus found the conic sections and was the first person to show that parabolas can be found by cutting a cone in a plane that is not parallel to the base. He did not "invent" the parabola. Accidentally, while trying to do duplicate the cube, he discovered conic sections and found the parabola.
A French man by the name of Pierre Fermat later on actually came up with the equation ay = x^2. Real life parabolas: If you kick a soccer ball it will arc up into the air and come down again following the path of a parabola. References:
http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Kim/emat6690/instructional%20unit/hyperbola/Hyperbola/Hyperbola.htm Understanding parabolas: y = ax 2 + bx + c◦
The role of 'a' ■If a> 0, the parabola opens upwards
■if a< 0, it opens downwards.
The axis of symmetry
The axis of symmetry is the line x = -b/2a
Picture of Standard form equation:
Axis of Symmetry from Standard Form: y= a(x-h)2+k
(h,k) is the vertex as you can see in the picture below Examples of parabolas: y = (x–1)² + 1
The parabola's vertex is the point (1,1) y = (x - 3)² + 4
(3,4) is the vertex. Hyperbola Def.
A plane curve having two separate parts or branches, formed when two cones that point toward one another are intersected by a plane that is parallel to the axes of the cones.
a^2+b^2=c^2 Hyperbola history: Again, the Greek get the credit for discovering this. Menaechmus first studied a special case of the hyperbola. This special case was xy = ab where the asymptotes are at right angles and this particular form of the hyperbola is called a rectangle hyperbola. (It makes more sense when you see it graphed.) Hyperbola in real life: Sonic Boom!!!!!!! (My favorite example)
As the plane moves faster than the speed of sound, you get a cone-like wave. Where the cone intersects the ground, it is an hyperbola.
Fun fact: The sonic boom hits every point on that curve at the same time. No sound is heard outside the curve. The hyperbola is known as the "Sonic Boom Curve." Hyperbola example: Draw the hyperbola given by This hyperbola opens right/left because it is in the form x - y.
a^2 = 9, b^2 = 4, c^2 = 9 + 4 = 20. Therefore, a = 3, b = 2 and c = 4.5