RWA#1: Unit M Concept 6a: Graphing hyperbolas and identyfing all parts

1. Mathematical Definition: The set of all points such the difference in the distance from two points (called the foci) is a constant. (Kirch)

2. Algebraically

   Graphically

   To classify the equation for a hyperbola, the standard form needs both terms squared and subtracted like so or. To find the points of the hyperbola we start with the center. (h,k) is our center. Then we look to find our ‘a’ value which is the distance from the center to the vertices and can be determined from the first denominator of the equation square rooted. The second denominator will be the ‘b’ value determining the distance from the center to the co-vertices. To find whether the graph is going to open left/right or up/down, we look to the first squared term. If x is squared first then the transverse axis will be horizontal meaning that the graph will go left/right. If y is squared first then the transverse axis will be vertical meaning that the graph will go up/down. The transverse and conjugate axis will always cross the center so if the transverse is on the x-value, then the conjugate is on the y-value, vice versa. To find the vertices, you can simply count from the center by the ‘a’ value, or add. It’s the same with the co-vertices with the ‘b’ value. Then there is the ‘c’ value that will give you the foci. For that we use c^2=a^2+b^2. For actually calculating the asymptotes in the slope-intercept form, we use if the graph is going left/right andor  if the graph is going up/down.

   A hyperbola looks like an ellipse that stretched so wide that it snapped in half and left it’s two halves going in opposite directions. Basically, the opposite-directions effect is caused by the foci which instead of being inside an ellipse are outside. Thus instead of the graph being a set of points for which the sum of distances from foci is a fixed constant, the difference of distances from the foci is a fixed constant. (Kirch) In this case, the constant is 2. Since the hyperbola is what farthest deviates from a circle, its eccentricity is greater than 1. Other parts of a hyperbola are the transverse and conjugate axis, two diagonal asymptotes, the center, vertices, co-vertices, and two branches. The transverse axis is determined by which term goes first in the equation or the vertices. The conjugate axis is determined as perpendicular to the transverse axis and also determined by the co-vertices. When drawing a rectangle from the transverse and conjugate axis, the two asymptotes go diagonally straight through it, never ending until it hits the graphs limits. All this is connected and crossing the center. The branches will be opening left/right or up/down, never touching the asymptotes and only the vertices. VIDEO BELOW

3.  RWA: Hyperbolic structures can be made out of straight beams, much easier than using curved beams that aren't as stable or strong. These structures make up water towers, cooling towers, and aesthetic features. The structure is very useful for radioactive cooling towers where at the bottom, the widening of the tower provides the area for installation of fill to cool circulated water. The narrowing of the middle helps support turbulent mixing when it then rises out of the widened opening.
 Many architects used the hyperbolic structure to make famous, looming buildings like the Church of Colonia Guell. Zarzuela, and the Kobe Port Tower. These towers were made my stacking sections of hyperboloids while supporting them with shape defining rings.
4. REFERENCES/WORKS CITED
https://www.youtube.com/watch?v=Z6cwpsDC_5A
http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/hyperbola-from-the-definition-geogebra-dynamic-worksheet
http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/all-the-conic-sections
http://www.pleacher.com/mp/mlessons/calculus/apphyper.html
http://en.wikipedia.org/wiki/Hyperboloid_structure