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Objectives: 1. Define a hyperbola 2. Write the equation of a hyperbola 3. Identify important characteristics of hyperbolas 4. Graph hyperbolas

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Hyperbola The set of all points for which the difference of the distances from two points is constant.

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Equation of a Hyperbola Centered on the Origin

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Characteristics of a Hyperbola Important Facts: The hyperbola bends toward the foci The positive term determines which way the hyperbola opens The distance between the foci is 2c The distance between the vertices is 2a The center is the midpoint between the foci and the midpoint between the vertices c 2 = a 2 + b 2

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Example #1 Show that the graph of the equation is a hyperbola. Graph it and its asymptotes. Find the equations of the asymptotes, and label the foci and the vertices.

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Example #1 Show that the graph of the equation is a hyperbola. Graph it and its asymptotes. Find the equations of the asymptotes, and label the foci and the vertices.

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Example #2 Graph the following hyperbola using a graphing calculator.

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Example #3A A. Find the equation of the hyperbola that has vertices at (2, 0) and (-2, 0) and passes through Then sketch its graph by using the asymptotes, and label the foci. With the vertices on the x-axis, this implies a = 2.

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Example #3A A. Find the equation of the hyperbola that has vertices at (2, 0) and (-2, 0) and passes through Then sketch its graph by using the asymptotes, and label the foci.

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Example #3B B. Find the equation of a hyperbola with y-intercepts at ±7 and an asymptote at With it intersecting the y-axis, this implies that a = 7. From the equation of the asymptote we get:

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Example #3C C. Find the equation of a hyperbola with foci at (±8, 0) and a vertex at

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Example #4 An airplane crashed and was heard by a park ranger and by a family camping in a park. The park ranger and the family are ¼ mile apart and the ranger heard the sound 1 second before the family. The speed of sound in air is approximately 1100 feet per second. Describe the possible locations of the plane crash. The family and the ranger are placed at opposite foci of a hyperbolic curve. The crash occurred closer to the ranger than the family so the crash occurred on the branch of the hyperbola closest to the ranger. Since sound travels at 1100 ft/sec, after 1 sec it will have traveled 1100 ft. This implies the crash was 1100 ft closer to the ranger than the family, which also means the vertices are 1100 ft apart. Since 1 mile has 5280 ft, ¼ a mile is 5280 ÷ 4 = 1320 ft, which is the distance between the foci.

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Example #4 Describe the possible locations of the plane crash. Distance between foci:1320 ft Distance between vertices:1100 ft The crash occurred somewhere on the left branch of the hyperbola.

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