Ch 4: The Hyperbola Objectives: Use the standard and general forms of the equation of a hyperbola Graph hyperbolas ©2003 Roy L. Gover (roygover@att.net)

Applications Hyperbolic Telescope The Hyperbola: Sonic Booms

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 graph of a hyperbola may look like this…

Definition …or like this:

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 rectangle

Definition The parts of a hyperbola are: the asymptotes

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

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

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

Try This Name the indicated parts of the hyperbola

Definition The distance from the center up to the vertex or down to the vertex is a units a For the “up & down” hyperbola, the transverse axis is vertical and the conjugate axis is horizontal Transverse Axis

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

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

a2 is always the first denominator Summary If x term is first, hyperbola opens left & right If y term is first, hyperbola opens up & down a2 is always the first denominator

Example Find the equation of the hyperbola that has foci at (2,5) and (-4,5) and a transverse axis 4 units long.

Try This Find the equation of the hyperbola that has foci at (4,0) and (-4,0) and the vertices are at (1,0) and (-1,0). Hint: first sketch the graph.

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

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

Try This 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

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