RWA# 1: Unit M Concept 5: Graphing Ellipses Given Equation

1. The mathematical definition of an ellipse is the set of all points such that the sum of the distance of two points, known as the foci, is a constant.(Unit M WPP)

Description of an Ellipses

 2. Describing an Ellipse
Algebraically: For an ellipses, the standard formula is (x-h)^2/a + (y-k)^2/b = 1. 
To solve one of these egg shaped ellipse, you  can quickly figure out the vertex you would take (h, k) only if it is in standard form of course. *An important side note would be the x-value stands for h and the the y-value stands for k. Once you have that done, you can identify the major axis if it is vertical or horizontal. In order to tell if it is horizontal the bigger denominator is under the x^2 term and vise versa for a vertical one.

The picture below shows you visually how one would be able to distinguish whether one will have a "fat or "skinny" ellipse.

https://blogger.googleusercontent.com/img/proxy/AVvXsEiF-p9cB7VeWY6ggFmgJ-C5PM4uYUGN7UKG7GuyElsL3JFmNzVG2O7ti0pB5WJet8DfmsQbIt0YdflirUcMC2Qvo06MDNdUysfkADiyje3mkSdT6EaXx3EX15rR27mWz6qD5euMbtltzjNsp8lr7UHx2qNcJvymTFiLOT_gdl2RdgK3Fg=

Graphically:
"The line through the foci intersects the ellipse at two points, the vertices.  The line segment joining the vertices is the major axis, and its midpoint is the center of the ellipse.  The line perpendicular to the major axis at the center intersects the ellipse at two points called the co-vertices (0, ± b).  The line segment that joins these points is the minor axis of the ellipse." (U.G)

https://blogger.googleusercontent.com/img/proxy/AVvXsEh-MrWwnCslVGn78w0MuvKBc7leqOVXGZDE7BVwVCJG4FnAurAzF7cW1AiuTStCOOlZbn-cIoeQfHBx3MzIJiYOyr78eXyR2ydPnx4g-Y2y6texok8eB7bkT2syZBmUW-3C7QkBYuv6cydCuJRso27xBEwXdsAPFNN7O7GY48Ctj4f5hxUh=

http://intmstat.com/plane-analytic-geometry/ellipsea.gif

Key Features:
The two graphs shown above contain a visual of all the key point of an ellipses. It includes a focus (f)which is c units away from the center. C is the distance from the center to the focus. It also contains a major and a minor axis which are labeled above on the graphs. The major axis is 2a length and the minor axis is 2b lengths. It has co-vertices which is just b units away. "The foci always lie on the major (longest) axis, spaced equally each side of the center. If the major axis and minor axis are the same length, the figure is a circle and both foci are at the center. Reshape the ellipse above and try to create this situation."( http://www.mathopenref.com/ellipsefoci.html) If you want to go in futher detail click here -> http://www.dummies.com/how-to/content/how-to-graph-an-ellipse.html

Foci and Eccentricity:
For an ellipses, the foci is normally placed between the vertices of the major axis. he ecccentricity of an ellipse is between 0 and 1 written like the following, 0<e<1. Found a nice website that will help understand the foci and eccentricity a little better by using a visual. http://www.mathopenref.com/ellipseeccentricity.html

Here's a video that will help you go step by step in solving and ellipses and tell you importants parts of it.

http://britton.disted.camosun.bc.ca/elliplanet_lg.JPG

 Real World Application: A real world application of an ellipses would be some buildings that actually are build like an ellipse if it were seen from a birds eye view. Architects use a center base to serve as the center point of the building.In the picture shown below you see a "skinny" ellipse that has its minor axis going from left to right. Its major axis is going from bottom to top. From the center point of the structure to the co-vertices the value is b.*one thing to keep in mind is that the focus always lies within the major axis. These components are taken into account when building an ellipse shaped structure. Like the following picture. The are used to appear pleasant to the eye. There are also some fins or airfoils that are made like that to cut through the wind. Things that are ellipses are found to have lower gravity in the center.
4. References:

• http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/emat6690/Insunit/ellipse/ellipse.html  
• http://intmstat.com/plane-analytic-geometry/ellipsea.gif
• http://www.dummies.com/how-to/content/how-to-graph-an-ellipse.html
• http://www.mathopenref.com/ellipseeccentricity.html
• http://www.mathopenref.com/ellipseeccentricity.html
• http://mathforum.org/library/drmath/view/62576.html 
•  http://britton.disted.camosun.bc.ca/elliplanet_lg.JPG