Warm up Find the equation in standard form of, Then give the coordinates of the center, the foci, and the vertices.

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Presentation transcript:

Warm up Find the equation in standard form of, Then give the coordinates of the center, the foci, and the vertices.

LESSON 10-4 HYPERBOLAS Objective: To use and determine that standard and general forms of the equation of hyperbola To graph hyperbolas

Hyperbola  The set of all points in a plane whose differences of the distances from two fixed points (foci) are constant.

Equation of a Hyperbola Co-vertices endpoints of conjugate axis Center: (h, k)

Equation of a Hyperbola Co-vertices endpoints of conjugate axis

Hyperbola b 2 = c 2 - a 2

Asymptotes: Lines that the branches approach but never intersect. or: open L/R: open U/D: equations: L/R:

Graph the following Hyperbola Center: (-1, 5) a = 4 in horiz direction b = 7 in vertical direction

Graph the following Hyperbola Center: (-1, 5) a = 4 b = 7 b 2 = c 2 - a = c = c 2 49= c 2 -16

Graph the following Hyperbola Asymptotes

Graph the following Hyperbola Asymptotes Center: (-1, 5) Vertices: (-5, 5) (3, 5) Co-Vertices: (-1, 12) (-1, -2) Length of Transverse axis: 8 Length of Conjugate axis: 14 Foci:

Equilateral and Rectangular Hyperbola equilateral hyperbolas are those in which a=b. rectangular hyperbolas are equilateral hyperbolas whose asymptotes are the coordinate axis. xy=c if c is positive the hyperbola will be in Quadrant I and III. if c is negative the hyperbola will be in Quadrant II and IV.

Eccentricity 0 1 the closer to 1 the less curve. add graphs the farther from 1 the more curve.

Ellipses vs. Hyperbolas c 2 = a 2 – b 2 the sum of the focal radii is 2a Equations: c 2 = a 2 + b 2 the difference of the focal radii is 2a Equations: L/R U/D

Ex. 1: Sketch the graph and find the foci of 16x 2 = 9y :

Ex. 2: Sketch the graph and find the foci of y 2 = 9x 2 + 9:

Practice Write the equation of the hyperbola with center at (-2,-4), a focus at (2,-4) and eccentricity 4/3.

Practice Find the coordinates of the center, foci, and vertices, and the equations of the asymptotes of the graph of Then graph the equations.

Sources Vernon Hills High School District 128. N.p., 2 Dec Web. 1 Feb "Algebra II Honors Period 4." Algonquin Regional High School. Algonquin Regional High School, n.d. Web. 1 Feb