RWA #1: Unit M: Concept 6: Hyperbola Conic Section in Real Life

All there is you need to know about Hyperbolas
1. Mathematical Definition. A hyperbola is "a curve where the distances of any point from a fixed point 9 (the focus)and a fixed straight line (the directrix) are always in the same ratio."(mathisfun.com)

2. Describing the Conic Section Algebraically

http://www.mathwarehouse.com/hyperbola/graph-equation-of-a-hyperbola.php

http://www.mathwarehouse.com/hyperbola/graph-equation-of-a-hyperbola.php

These two pictures show the two formulas used for a hyperbola. The reason there are two different formulas is because the way each hyperbola branches out. This means that if the equation begins with x it will open up on the x- axis. You will also know that it is the horizontal transverse axis. If the equation begins with y it will open on the y-axis and you will also know that it is the vertical transverse axis. The different parts of this conic section on this axis are vertices, foci, and center. The conjugate axis includes co-vertices. The conjugate axis is the direction in which the hyperbola opens. Asymptotes are also very important they are shown as y=mx+b. The formula to find the asymptotes for a transeverse axis which is horizontal us y=k+ or - b/a (x-h). The formula to find the asymptotes for a transverse axis which is vertical is y=k + or - a/b (x-h).

Graphically:

http://www.sparknotes.com/math/precalc/conicsections/section1.rhtml

To find the center which is (h,k) they will be included in the formula so they are paired off with x and y. X is always with h and y is always with k. To find the vertices you use what you have already which is the center. For example if the hyperbola y is first then you find the center and then use the x from the center to find the vertices which is the number for a up and the number for a down. The same with the co-vertices except you go left and right. A and B are found when you take the square root of the denominators. If y is first a squared is underneath it and b squared is underneath x or vice versa. To find the asymptotes you use the formula given in the above paragraph.

Focus and Eccentricity

http://www.youtube.com/watch?v=S0Fd2Tg2v7M     (Khan Academy)

This video explained how an ellipse and a hyperbola are similar even with their foci. The eccentricity of a hyperbola should be 1 or above.The closer the eccentricity is to one the curves of the hyperbola will make it look sharper or more pointy. The farther away the eccentricity is to one it will make the curves of the hyperbola look straighter, which means it will look fatter. A smaller foci means a small width and a larger foci means a larger width.

3. Hyperbolas in real life

http://www.pleacher.com/mp/mlessons/calculus/apphyper.html

Hyperbolas can be found all over the real world for example a common household lamp can cast a shadow that is a hyperbola. Also in architecture like the Dulles Airport which is shaped like a hyperbola. The one i picked is a nuclear power plant's cooling tower. The reason that this is the standard shape for all cooling towers is because it needs to withstand high winds and be built with little material. When the cooling tower is in the shape of a hyperbola it is easier for the building to take the damage of high winds. This shape also is very inexpensive to build with steel beams and concrete.
As you can see the cooling tower looks like a cylinder. Well when you take a cylinder by each side and  push one side forward and one side back you will create a hyperbola. Hyperbolas can also be found with jets. As a jet breaks the sound barrier it releases a sonic boom. When the boom is released a cloud is formed and it looks like a cone, when it comes into contact with the ground then it becomes a hyperbola.

4. References
Vertical and Horizontal Transverse Axis of Hyperbola:
http://www.mathwarehouse.com/hyperbola/graph-equation-of-a-hyperbola.php
http://www.mathwarehouse.com/hyperbola/graph-equation-of-a-hyperbola.php

Different Conic Sections:

http://www.sparknotes.com/math/precalc/conicsections/section1.rhtml

Focus and Eccentricity: Khan Academy
http://www.youtube.com/watch?v=S0Fd2Tg2v7M

Hyperbola in Real World:
http://www.pleacher.com/mp/mlessons/calculus/apphyper.html