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| (http://www.tpub.com/math2/Job%202_files/image777.jpg)
Standard Form
An ellipse has two standard forms. If you look at the image above you can see that the first one has a^2 under the x. This indicates that the graph will be a fat graph and that your major axis is equal to y and the minor axis will be equal to x. This is because the graph will have a constant y while the x changes to make the graph extend horizontally. The second equation has the a^2 under the y. This means that your graph will be skinny and that your major axis will be equal to x and your minor axis will be equal to y. This is due to the fact that your x will remain constant as you move up and down, stretching it vertically.
Center
The ellipse has a center just like every other conic. The center will always derive from the x and y binomials. If you look at the picture above, you see that x is paired with an h and y is paired with a k. Those are the numbers that you will use to write your center. The center is opposite of the numbers represented in the binomials. That means that if h is positive in the parenthesis then it will be negative in the ordered pair and vice versa. It is an easy concept to understand and you want to make sure that you do not forget to switch the signs. Remember that h will always be your x coordinate and k will always be your y coordinate.
A and B
When you look at the image above you see a^2 and b^2. These denominators will help you determine what your vertices and co-vertices will be and also help you determine your c and foci. The square root of the larger denominator will be your a. The a is the distance between your center and each of your vertices. The whole major axis is a distance of 2a. (vertex to vertex). This will allow you to figure out what your vertices are. You will move a spaces up and down or left and right depending on your graphs shape. The b is similar. You will find b by taking the square root of the smaller denominator. The b is the distance from your center to each of your co-vertices. That is either up and down or left and right from the center depending on the graphs shape. The length of the minor axis is put in terms of 2b (co-vertex to co-vertex). The b will also let you indicate what your co-vertices are by counting from the center on the minor axis that many spaces. Besides this though is the fact that they allow you to find your c.
C
The c is an important part of the graph. The c allows you to figure out your eccentricity and your foci. Your c is found by taking a^2 and subtracting b^2 to get c^2. Then you take the square root of c and simplify if possible. That answer will become your c. The c is then later used to find your foci and eccentricity. The equation is also shown below.
Foci
The foci are two ordered pairs that are derived from the center and c. The c is added and subtracted from the minor axis term. If the minor axis is y, then it will be added/subtracted from that term. The foci are two ordered pairs that lie on the major axis within the ellipse. They are a key part of the graph because they indicate also whether the graph will be stretched or shrunk. They are on the major axis and the major axis tells you whether the graph will be stretched or shrunk.
Eccentricity
According to the SSS Packet, the eccentricity is a measure of how much the conic section deviates from being circular. To find the eccentricity of an ellipse you take c and divide it by a. This will give you a number that is less than one, but greater than zero. It is the distance from your center to your focus divided by the distance from the center to a vertex. This is shown in the image below.
 (http://freemathdictionary.com/wp-content/uploads/Eccentricity-.gif)
Major Axis
The major axis is the axis that dominates the graph. The major axis is represented by a solid line on most graphs. It contains the center, the vertices, and the foci. The major axis is seen as y=k for ellipses whose x has the larger denominator in standard form. If the y is the one who has the larger denominator then the major axis is seen as x=h. These are the lines that tell you if the graph is fat or skinny (stretched horizontally or shrunk vertically). This is an important part and will match your a, the unchanging term in your vertices, and your foci.
Minor Axis
The minor axis is the axis that acts as a guide to size. It crosses through the center, but nothing lies on it. It has the co-vertices as endpoints. It is the b from your standard equation. It is seen as x=h if the x has the larger denominator. If the y has the larger denominator then it is y=k. This is the simple one. It is not too difficult to understand or figure out. If you look at the image above, the minor axis is pretty much an empty line. In most graphs it will be seen as a dotted line.
Vertices and Co-Vertices
The vertices and the co-vertices are probably some of the easiest points to find on a graph besides the center. The vertices are a distance from the center and the co-vertices are b distance from the center. Their constant coordinate is the one that matches what axis they lie on. If is the vertex, the constant variable will match that of the major axis and if it is the co-vertex, the constant variable will match that of the minor axis. They are your endpoints for both your major and minor axis. They are your ellipse shape guidelines.
This is just another example of what an ellipse looks like on the graph. It is an example of a skinny graph because the foci have the same x and a different y, meaning that the x is the major axis.
Real World Application
An example of an ellipse in the real world is the whisper chamber in the capital building. The whisper chamber is found in statuary hall in the capital building. It is an ellipse. This room has two foci that lie in a straight line from each other on the opposite side of the room. If you stand on one of these foci and your friend on the other and you whisper something to your friend, they will be able to hear you. If you take a single step off of the focus then the sound will no longer reach your friend. This is because of he reflective properties of an ellipse. This property is an interesting one. If you drew a line in that room those two spots would be on the same line, meaning that it is the major axis. The center would be the middle ground of those two points. The reason that the whispers travel is due to the reflection off of the walls in the room. No matter what way they bounce off the wall the distance is the same. This distance is a constant. The whispers bounce off in all directions, but each one is the same.
The whispering room in the capital room is a great example of a real life ellipse because it puts into action the distance concept. The whispering spots are an equal distance from each other no matter where on the wall the sound reflects off of. This allows you to hear your friends and anyone else who might be standing on that spot at the same time that you are. It is not only found in this room though. There are other buildings in the world that do the same thing. The capital building is just a good example of where we find it in our capital. Having been in the room and testing this myself, it is very interesting to find out about. The reflection theory is present and the room is shaped like an ellipse. The foci are the gold plates on the floor and the distance is in fact the same no matter the direction of the reflection. For more information, watch the video below and read the following websites.
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https://www.teachervision.com/math/resource/5980.html
http://www.pleacher.com/mp/mlessons/calculus/appellip.html
References
https://www.teachervision.com/math/resource/5980.html
http://www.pleacher.com/mp/mlessons/calculus/appellip.htmlhttp://moodle.oakland.k12.mi.us/os/pluginfile.php/40772/mod_book/chapter/454/Conic_Sections/images/ellipses_ex1-3.png
http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/ellipse-drawn-from-definition-geogebra-dynamic-worksheet
http://www.tpub.com/math2/Job%202_files/image777.jpg
http://www.mathwarehouse.com/ellipse/images/formul-focus.gif
http://freemathdictionary.com/wp-content/uploads/Eccentricity-.gif
http://hotmath.com/hotmath_help/topics/ellipse/ellipse-image012.gif
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