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Copy of 9-3 Hyperbola Graphing
Transcript of Copy of 9-3 Hyperbola Graphing
Solution for Example 2
x^2 - 9y^2 + 36y - 72 = 0
The word "hyperbola" comes from the Greek word meaning "over-throw". Hyperbolae were discovered by Menaechmus, during his research on doubling the cube.
A hyperbola has two disconnected curves called branches that continue to infinity. Various parts of a hyperbola include the vertices (the two points on opposite branches of the hyperbola that are closest to each other), the asymptotes (lines that the branches never touch but intersect the hyperbolas center), the major axis (the distance between the vertices), and the foci (points such that the difference of the distance between any point on the graph and the points themselves is constant). The hyperbola has mirror symmetry about the principle axis.
When an airplane flies at the speed of sound, it creates a sonic boom
the sonic boom "cloud" is visible due to humidity in the air
When the sonic boom hits the ground, it creates a cone shape
Voyager II- Space Travel
NASA launched the Voyager II in August 1977 to prove the concept of "gravity assist"
Real World Application
9-3 Hyperbola Graphing
By: Shannon Chen
may increase or decrease a spacecraft's speed through the addition or subtraction of momentum
gravitational force of planets such as Jupiter allow the Voyager to fall toward Jupiter but the speed and momentum of the spacecraft was sufficient enough for the spaecraft to curve and fly away from Jupiter
in this picture the cone shaped sonic boom
creates a U-shaped curve on the ground
anyone along the curve would feel the boom
the planes sound waves travel in hyperbolic motions
The term itself was believed to have been coined by Apollonius of Perga, during his extensive work with conic sections
(x+3)^2/144 - (y-2)^2/25 =1
a.) Find the standard form of the equation of the hyperbola, b.) find the center, vertices, foci, and asymptotes of the hyperbola, c.) sketch the hyperbola
Find the center, vertices, foci, and asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid.
~center: (-3,2) ~ a=12 b=5 ~V1 (9,2); V2 (-15,2)
~F1 (-16,2); F2 (10,2) ~ asymptotes:
Solution for Example 1
a.) x^2/36 - (y-2)^2/4 = 1
b.) C (0,2) ; V1 (-6,2) V2 (6,2)
F1 (-2sq.10, 2) F2 (2sq.10, 2)
c.) graph on board
Rutherford used hyperbolas to determine the maximum size of the nucleus by observing the maximum possible deflection
The diffraction of light also causes the formation of a hyperbola (a rainbow)
graph on board
When the sonic boom is felt and heard at the same time, then the aircraft flies faster that the speed of sound.
Sonic Boom Application Question
Assume that you are standing 500 feet away from the tip of the hyperbolic curve.
Find the point V.
History of the Hyperbola
Hyperbolas are present in everyday life, even when it is least expected. Some examples are nuclear cooling towers, sundials, and shadows emitted by lamps