Definition A hyperbola is the set of all points such that the difference of the distance from two given points called foci is constant.

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Presentation transcript:

Definition A hyperbola is the set of all points such that the difference of the distance from two given points called foci is constant

Definition The parts of a hyperbola are: transverse axis

Definition The parts of a hyperbola are: conjugate axis

Definition The parts of a hyperbola are: center

Definition The parts of a hyperbola are: vertices

Definition The parts of a hyperbola are: foci

Definition The parts of a hyperbola are: the asymptotes

Definition The distance from the center to each vertex is a units a The transverse axis is 2 a units long 2a2a

Definition The distance from the center to the rectangle along the conjugate axis is b units b 2b2b The length of the conjugate axis is 2 b units

Definition The distance from the center to each focus is c units where c

Sketch the graph of the hyperbola What are the coordinates of the foci? What are the coordinates of the vertices? What are the equations of the asymptotes?

How do get the hyperbola into an up-down position? switch x and y identify vertices, foci, asymptotes for:

Definition where ( h, k ) is the center Standard equations:

Definition The equations of the asymptotes are: for a hyperbola that opens left & right

Definition The equations of the asymptotes are: for a hyperbola that opens up & down

Summary Vertices and foci are always on the transverse axis Distance from the center to each vertex is a units Distance from center to each focus is c units where

Summary If x term is positive, hyperbola opens left & right If y term is positive, hyperbola opens up & down a 2 is always the positive denominator

Find the coordinates of the center, foci, and vertices, and the equations of the asymptotes for the graph of : then graph the hyperbola. Hint: re-write in standard form Example

Solution Center: (-3,2) Foci: (-3±,2) Vertices: (-2,2), (-4,2) Asymptotes:

Example Find the coordinates of the center, foci, and vertices, and the equations of the asymptotes for the graph of : then graph the hyperbola.

Solution Center: (-4,2) Foci: (-4,2± ) Vertices: (-4,-1), (-4,5) Asymptotes: