♕ The Math Analysis Elite ♕: RWA# 1: Unit M Concept 5: Graphing Ellipses Given Equations and Identifying All Parts

1. Mathematical Definition:
Ellipses - "The set of all points such that the sum of the distance from two points is a constant." (Kirch)

2. Description:
Algebraically: Ellipses can take on two different forms, either "fat" or "skinny". These two deviations are reflected in the algebraic representations. "a^2” is located beneath the major axis term.

(Pauls Online Notes)

Graphically: Ellipses have an eccentricity of less than one. Due to the fact that it is greater than 0, it deviates from being a perfect circle. This leaves a graph looking similar to a circle, however it is elongated on two ends.

(Kirch)

(google)

Key Features: Ellipses contain 12 key features. The first feature would be getting the equation in standard form. Standard Form, as seen above, allows problem solver to acquire a great deal of info from the ellipse. They are apple to ascertain the center, whether it will be skinny or fat, and even "a" and "b" values (these later aid in finding other features such as the major and minor axis, as well as the foci and eccentricity.). You find standard form through the action of completing the square; not only do you have to complete the square, but you also have to set the equation equal to 1. With the standard form equation, you can determine whether the graph will be skinny or fat. If the bigger denominator is below the "x" term, the ellipse will be fat. However, if the larger denominator is below the "y" term, it will be a skinny ellipse. This can easily be seen on a graph as the ellipse is obviously skinny or fat. One of the easiest features that can be found for an ellipse is the center! This is done by taking the "x" and "y" terms from the standard form equation and turning them into an ordered pair. While doing this, it is important to remember that you need to flip the positives and negatives! The center is even easier to find on the graph! Needless to say, it is at the center of the ellipse.

With the center, you can go on to find the major and minor axi (axises? multiple axis word). The major or minor axis will be set equal to either x or y. If the larger denominator is under the "x" value, you will set the major axis as "y=___" while if the larger denominator is under the "y" value, you will set the major axis as "x=___". Another one of the easier parts of solving an ellipse comes with "a" and "b". To find "a", you have to take the square root of the largest denominator. You can spot this on a graph by looking at the center and the points, then go on to determine which axis is which.For the "b" value, take the square root of the other denominator left. To find the two major vertices, take the "x" or "y"value from the center (depending on your major axis); then use the "a" value to move up/down or left/right from the center to find two points. For the 2 co-vertices take the "x" or "y"value from the center (depending on your major axis); then use the "b" value to move up/down or left/right from the center to find two points. These are easy to spot as they are simply points on a graph. The "c" value is more complicated to come by. For this, you have to use the equation "a^2-b²=c^2" to find the "c" value. Once the "c" value is found you create a new ordered pair of foci. For this, you have to use the center and manipulate it. You take the minor axis value then add and subtract the "c" value (in square root form), with the major axis value completing the pair. The foci effect the shape of the conic section as they make it bigger or smaller. To find the eccentricity, use the equation"c/a" and get a decimal as your answer!

The video below may help further your understanding.

3. Real World Application:
Ellipses are very common throughout every day life. One of the most notable example is that of our solar system. Prior to the discovery that planets moved in an elliptical manner around the sun, Ancient Greeks believed that the solar system was based upon a circular rotation. This was later changed as philosophers acknowledged that the sun acted as a foci around which the planets could revolve around. (Occurrence of the Conics)
In this example, the planets would be the line connecting the the points along the path. The farthest points or vertices would be that of either winter or summer depending on the time of year. The between months would dictate the location of the co-vertices. Lastly, the foci would reflect the sun and another similar point to the system.