Examples

Example 1

Solution for Example 2

Example 2

x^2 - 9y^2 + 36y - 72 = 0

Part 1

The word "hyperbola" comes from the Greek word meaning "over-throw". Hyperbolae were discovered by Menaechmus, during his research on doubling the cube.

Part 3

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.

Sonic Boom

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**

Jamie Park

Will Abele

Kris Kaya

Jamie Park

Will Abele

Kris Kaya

Gravity Assist

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:

y-2=(5/12)(x+3)

y-2=(-5/12)(x+3)

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)

asymptotes: y-2=(1/3)(x-0)

y-2=(-1/3)(x-0)

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.

V

V

You

500 ft

Part 2

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