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Understanding Hyperbolas: Center, Vertices, Foci, and Graphing

In this session, we explore hyperbolas, focusing on completing the square and determining eccentricity. We'll prove that the graphs of specific quadratic equations represent hyperbolas and find their centers, vertices, and foci. Using two examples, we demonstrate how to reformulate equations and graph hyperbolas by hand. Key topics include eccentricity, the semi-transverse axis, and the semi-conjugate axis, along with homework exercises and quiz preparation. Join us for an engaging look at these fascinating conics!

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Understanding Hyperbolas: Center, Vertices, Foci, and Graphing

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Presentation Transcript

  1. Today in Precalculus • Go over homework • Notes: Hyperbolas Completing the square Eccentricity • Homework

  2. Hyperbolas Prove that the graph of 9x2 – 4y2 – 18x + 8y + 41 = 0 is an hyperbola. Find the center, vertices and foci. Then graph the hyperbola by hand. 9x2 – 18x – 4y2 + 8y = -41 9(x2 – 2x) – 4(y2 – 2y) = -41 9(x2 – 2x + 1) – 4(y2 – 2y + 1) = -41 + 9 - 4 9(x – 1)2 – 4(y – 1)2 = -36

  3. Center: (1, 1) a = ±3 Vertices: (1, -2), (1, 4) c2 = 9 +4 = 13 c = ±3.6 Foci: (1, -2.6), (1,4.6 ) Pts on conjugate axis b = ±2 (-1,1), (3,1)

  4. Example 2 Prove that the graph of 9y2 – 25x2 + 72y – 100x +269 = 0 is an hyperbola. Find the center, vertices and foci. Then graph the hyperbola by hand. 9y2 + 72y – 25x2 – 100x = -269 9(y2 + 8y) – 25(x2 + 4x) = - 269 9(y2 + 8y + 16) – 25(x2 + 4x + 4) = -269 +144 -100 9(y + 4)2 – 25(x + 2)2 = -225

  5. Center: (-2, -4) a = ± 3 Vertices: (-5, -4), (1, -4) c2 = 9 + 25 =34 c = ±5.8 Foci: (-7.8, -4), (3.8, -4) Pts on conjugate axis b = ±5 (-2, -9), (-2, 1)

  6. Eccentricity Where a is the semitranvserse axis and b is the semiconjugate axis. Example a: Example b:

  7. Homework Page 664: 47-50 Ellipse and Hyperbola Quiz: Tuesday, February 26

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