2. The standard formula of an ellipse is (x-h)^2/a + (y-k)^2/b = 1. On a graph, an ellipse is depicted as an oval, or a squashed circle. In other words, it is stretched out. The key features of an ellipse are the standard form, the center, vertices, co-vertices, two foci, major axis, minor axis, a, b, and c, eccentricity, and if it is seen as "skinny" or "fat". You can find these graphically or algebraically.
Lets start off with algebraically. By looking at the standard form of an ellipse, you can easily find the center. The x value stands for "h" and the y value stand for "k". After you find your center, you can find your major axis by looking at the denominator. If the denominator is bigger under the first term (the x), that is your major axis. This also means that your ellipse will stretch horizontally, being "fat". If the bigger denominator is under the y, then then your graph will stretch vertically, being "skinny". Once you have your major axis, you can find your minor axis because it would be opposite. The smaller number x is your minor axis and horizontal. If the smaller number is under y then it is vertical. For major and minor, if one is vertical, you know the equation is x= h is. If it is horizontal, y=k.You can find your a and b from the standard formula too. The bigger denominator is the a^2 term so you have to square root it to find your a. The smaller denominator is your b^2 term and you also have to square root it to find your b. After that, you can find your c term by plugging your "a" and "b" into the formula, a^2+b^2=c^2. To find the vertices, you add and subtract "a" from your center. The major axis stays the same throughout. To find the co-vertices, you add and subtract "b" from your center, the minor axis staying the same. To find the foci, you add "c" to the value that's changing. It's on the major axis. If major axis is x=, you add "c" to your "k" (y value of center). It is plus or minus because there are two vertices. If the foci is closer to the center, the ellipse is more circular. If it is farther, it is more stretched out. Eccentricity is a measure of how much the conic section deviates from being circular. If it is closer to 1, is it more stretched out. If it is closer to 0, it is more in the circular form. The eccentricity is between 0 and 1. To find the eccentricity, you use the formula c/a and round to the thousand's place.
To identify an ellipse graphically, it is simple as well. The major axis is the longer diameter of the ellipse, and is seen with a straight line (no dashes). The minor axis is the axis intersecting the major axis, it is shorter, and seen with a dashed line. The vertices are the points that lie on the major axis. The co-vertice are the points that connect the minor axis, at the end. With the vertices and co-vertices, the ellipse can be drawn. The center of the ellipse is where the major and minor axis intersect. If the ellipse is seen more horizontally, then it is "fat". If it is seen as more vertically stretched, it is "skinny" You can determine a by counting how many points are away from the center from each vertices on the major axis. Same goes with how to find b, you can count how many points are away from the center of the co-vertices on the minor axis. The foci is within the vertices of the major axis. The closer the foci to the center, the more it is in circular form. The farther the foci, the more it deviates from being a circle. If the foci is close to the center, the eccentricity is close to 0. If foci is far from center, eccentricity is big.
Here is a website that shows how to graph an ellipse!
Here is an ideal graph of an ellipse with some of the points I talked about.
This is a cool video on how to find the foci!
3. A real world application of an ellipse is that it is used in lithotripsy. "One important property of the ellipse is its reflective property. If you think of an ellipse as being made from a reflective material then a light ray emitted from one focus will reflect off the ellipse and pass through the second focus. This is also true not only for light rays, but also for other forms of energy, including shockwaves. Shockwaves generated at one focus will reflect off the ellipse and pass through the second focus."(http://mathcentral.uregina.ca/beyond/articles/Lithotripsy/lithotripsy1.html).
Therefore, it has been used my medical experts to create a device that helps kidney stones and gallstones. A lithtripter uses shockwaves to SHATTER a kidney stone or gallstone. Patients with kidney stones or gallstones can be saved because of the shape of an ellipse because it reflects shockwaves. HOW COOL IS THAT?! MATH SAVES LIVES AND IT WILL CONTINUE TO. :D