A hyperbola is a set of all points such that the difference in the distance from two points is a constant.
In a hyperbolas it matters which term (x or y) comes first in the equation because the equation of the hyperbola contains a minus instead of a plus like the ellipse. Therefore, the graph will look different based on which term (x or y) comes first.
The equation of the hyperbola can either be (x-h)^2/a^2 all minus (-!) (y-k)^2/b^2=1 or (y-k)^2/a^2 all minus (x-h)^2/b^2. Notice that the bottom terms (a and b) are not switched around and instead the a stays with the first term always and the b with the second term always. Also notice that when the x and y terms are switched the center values (h,k) also switch, meaning that the h value always matches the x value in the point and the k value always matches the y value in the point of the graph. DO NOT mix those two values when writing out the center for the graph.
http://t0.gstatic.com/images?q=tbn:ANd9GcTAP1wXzUwZiM2d8qshPZqrjVmj3GTa2FeHAOLsXfTbpKlsz5nr:mysite.du.edu/~jcalvert/math/hype1.gifWhen x comes first in the equation you should already know that the graph will have the hyperbola going left and right. Also, this means that the transverse axis (main axis) is horizontal and for the vertices the x changes value and the y stays the same ((x,y) from the point of the center, which you get from (h,k)). Meanwhile, when y comes first in the equation the graph will have the hyperbola going up and down and the transverse axis will be vertical, meaning the vertices x will stay the same but the y value will change.
After figuring out these simple connections between the algebraic and graphic hyperbolas, it will be simple to discover other hyperbola elements such as the conjugate axis, the co-vertices and the foci. Remember that the foci of hyperbolas are outside the box drawn in the graph.
To have a better visual with the information given above please do not be a cheesebucket and check out video below:
REAL LIFE APPLICATIONS:
5. Cooling Towers of Nuclear Reactors - The hyperboloid is the design standard for all nuclear cooling towers. It is structurally sound and can be built with straight steel beams. When designing these cooling towers, engineers are faced with two problems: (1) the structutre must be able to withstand high winds and (2) they should be built with as little material as possible. The hyperbolic form solves both of these problems. For a given diameter and height of a tower and a given strength, this shape requires less material than any other form. A 500 foot tower can be made of a reinforced concrete shell only six or eight inches wide. See the pictures below (this nuclear power plant is located in Indiana).
6. Stones in a Lake - When two stones are thrown simultaneously into a pool of still water, ripples move outward in concentric circles. These circles intersect in points which form a curve known as the hyperbola.
Source:
http://www-prod.pen.k12.va.us/Div/Winche...
http://t0.gstatic.com/images?q=tbn:ANd9GcQTbwo-QERHjE7sHXVRwPD7K_YE5lnNO3FSv2Nj0zuqhowGicuBSQ:education.ti.com/en/timath/~/media/Images/Activities/US/Math/Algebra%2520II/Exploring%2520Hyperbolas/ExploringHyperbolas3.jpg
To understand the picture above of the real world application for hyperbolas go ahead and check out the link below:
press here: http://www.pleacher.com/mp/mlessons/calculus/apphyper.html
Work Cited:
http://t0.gstatic.com/images?q=tbn:ANd9GcQTbwo-QERHjE7sHXVRwPD7K_YE5lnNO3FSv2Nj0zuqhowGicuBSQ:education.ti.com/en/timath/~/media/Images/Activities/US/Math/Algebra%2520II/Exploring%2520Hyperbolas/ExploringHyperbolas3.jpg
http://www.pleacher.com/mp/mlessons/calculus/apphyper.html
<iframe width="500" height="375" src="http://www.schooltube.com/embed_force/2eec3f755fdc4a84a617/" frameborder="0" allowfullscreen="allowfullscreen" mozallowfullscreen="mozallowfullscreen" webkitallowfullscreen="webkitallowfullscreen"></iframe>
http://kirchmathanalysis.blogspot.com/p/unit-m.html
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