10.3 Hyperbolas. Circle Ellipse Parabola Hyperbola Conic Sections See video!

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Presentation transcript:

10.3 Hyperbolas

Circle Ellipse Parabola Hyperbola Conic Sections See video!

Where do hyperbolas occur?

Hyperbolas Hyperbola: set of all points such that the difference of the distances from any point to the foci is constant. Difference of the distances: d 2 – d 1 = constant vertices The transverse axis is the line segment joining the vertices. The midpoint of the transverse axis is the center of the hyperbola.. asymptotes d1d1 d1d1 d2d2 d2d2

Standard Equation of a Hyperbola (Center at Origin) This is the equation if the transverse axis is horizontal. (–a, 0)(a, 0) (0, b) (0, –b)

Standard Equation of a Hyperbola (Center at Origin) This is the equation if the transverse axis is vertical. (0, –a) (0, a) (b, 0)(–b, 0)

How do you graph a hyperbola? To graph a hyperbola, you need to know the center, the vertices, the fundamental rectangle, and the asymptotes. Draw a rectangle using +a and +b as the sides... (–4,0)(4, 0) (0, 3) (0,-3) a = 4 b = 3 The asymptotes intersect at the center of the hyperbola and pass through the corners of the fundamental rectangle Example: Graph the hyperbola Draw the asymptotes (diagonals of rectangle)... Draw the hyperbola...

Example: Write the equation in standard form of 4x 2 – 16y 2 = 64. Find the vertices and then graph the hyperbola. Get the equation in standard form (make it equal to 1): 4x 2 – 16y 2 = (–4,0)(4, 0) (0, 2) (0,-2) That means a = 4 b = 2 Vertices: Simplify...

Standard Equations for Translated Hyperbolas