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10.5 Hyperbolas

10.5 Hyperbolas. What you should learn:. Goal. 1. Graph and write equations of Hyperbolas. Goal. 2. Identify the Vertices and Foci of the hyperbola. Identify the Foci and Asymptotes. Goal. 3. 10.5 Hyperbolas. Hyperbolas.

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10.5 Hyperbolas

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  1. 10.5 Hyperbolas Whatyou should learn: Goal 1 Graph and write equations of Hyperbolas. Goal 2 Identify the Vertices and Foci of the hyperbola. Identify the Foci and Asymptotes. Goal 3 10.5 Hyperbolas

  2. Hyperbolas • Like an ellipse but instead of the sum of distances it is the difference • A hyperbola is the set of all points P such that the differences from P to two fixed points, called foci, is constant 10.5 Hyperbolas

  3. Hyperbolas • The line thru the foci intersects the hyperbola at two points (the vertices) • The line segment joining the vertices is the transverse axis, and it’s midpoint is the center of the hyperbola. • Has 2 branches and 2 asymptotes • The asymptotes contain the diagonals of a rectangle centered at the hyperbolas center 10.5 Hyperbolas

  4. Standard Form of Hyperbola w/ center at origin Foci lie on transverse axis, c units from the center c2 = a2+b2 10.5 Hyperbolas

  5. Asymptotes (0,b) Vertex (a,0) Vertex (-a,0) Focus Focus (0,-b) This is an example of a Horizontal Transverse axis (a, the biggest number, is under the x2 term with the minus before the y) 10.5 Hyperbolas

  6. Vertical Transverse axis 10.5 Hyperbolas

  7. Example) Graph the equation. 36 36 36 • a = 3 b = 2 • because term is positive, the transverse axis is horizontal & vertices are • (-3,0) & (3,0) 10.5 Hyperbolas

  8. Example) Graph 4x2 – 9y2 = 36 • Draw a rectangle centered at the origin. • Draw asymptotes. • Draw hyperbola. 10.5 Hyperbolas

  9. 10.5 Hyperbolas

  10. Write the equation of a hyperbola with foci (0,-3) & (0,3) and vertices (0,-2) & (0,2). • Transverse axis is Vertical because foci & vertices lie on the y-axis • Center is the origin because foci & vertices are equidistant from the origin • Since c = 3 & a = 2, c2 = b2 + a2 • 9 = b2 + 4 • 5 = b2 • +/-√5 = b 10.5 Hyperbolas

  11. Reflection on the Section How are the definitions of ellipse and hyperbola alike? How are they different? Both involve all points a certain distance from 2 foci; For an ellipse, the sum of the distances is constant; for a hyperbola, the difference is constant. assignment Page 618 # 15 – 63 odd 10.5 Hyperbolas

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