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LC.02.2 - The Hyperbola (Algebraic Perspective)

LC.02.2 - The Hyperbola (Algebraic Perspective). MCR3U - Santowski. (A) Review. The standard equation for a hyperbola is x 2 /a 2 - y 2 /b 2 = 1 (where the hyperbola opens left/right/along the x-axis and the foci on the x-axis and where the transverse (major) axis is on the x-axis)

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LC.02.2 - The Hyperbola (Algebraic Perspective)

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  1. LC.02.2 - The Hyperbola (Algebraic Perspective) MCR3U - Santowski

  2. (A) Review • The standard equation for a hyperbola is x2/a2 - y2/b2 = 1 (where the hyperbola opens left/right/along the x-axis and the foci on the x-axis and where the transverse (major) axis is on the x-axis) • Alternatively, if the foci are on the y-axis, and the transverse (major) axis is on the y-axis, the hyperbola opens up/down/along the y-axis, and the equation becomes x2/b2 - y2/a2 = -1 • In a hyperbola, the “minor axis” is referred to as the conjugate axis, but is not really a part of the graph of the hyperbola • The intercepts of our hyperbola are at +a (opening L/R) • The vertices of the hyperbola are at +a and the length of the transverse (major) axis is 2a • The domain is -a>x>a and range is yER for hyperbola opening L/R • The two foci are located at (+c,0) for opening L/R • The asymptotes of the hyperbola are at y = (+b/a)x for opening L/R • NEW POINT  the foci are related to the values of a and b by the relationship that c2 = a2 + b2

  3. (B) Translating Hyperbolas • So far, we have considered hyperbolas from a geometric perspective |PF1 - PF2| = 2a and we have centered the hyperbolas at (0,0) • Now, if the hyperbola were translated left, right, up, or down, then we make the following adjustment on the equation: • So now our centrally located hyperbola has been moved to a new center at (h,k)

  4. (C) Translating Hyperbolas – An Example • Given the hyperbola determine the center, the vertices, the foci, the intercepts and the asymptotes. Then graph • The center is clearly at (3,-4)  so our hyperbola was translated from being centered at (0,0) by moving right 3 and down 4  so all major points and features must also have been translated R3 and D4 • The transverse axis is on the x-axis so the hyperbola opens L/R • The value of a = 4 and b = 5 • So the original vertices were (+4,0)  the new vertices are (-1,-4) and (7,-4) • The endpoints of the “minor” axis were (0,+5)  these have now moved to (3,1), (3,-9) • The original foci were at (52 + 42) = +6.4  so at (+6.4,0) which have now moved to (-3.4,-4) and (9.4,-4) • The asymptotes used to be the lines y = +1.25x, which have now moved to y = +1.25(x – 3) - 4

  5. (C) Translating Hyperbolas – The Intercepts So no y-intercepts

  6. (C) Translating Hyperbolas – The Graph

  7. (D) In-Class Examples • Ex 1. Graph and find the equation of the hyperbola whose transverse axis has a length of 16 and whose conjugate axis has a length of 10 units. Its center is at (2,-3) and the transverse axis is parallel to the y axis (so it opens U/D) • So 2a = 16, so a = 8 • And 2b = 10, thus b = 5 • The asymptotes were at y = (+8/5)x (Since the hyperbola opens U/D, the asymptotes are at y = (+a/b)x) • And c2 = a2 + b2 = 64 + 25 = 89  c = +9.4 • Therefore our non-translated points are (0,+8), (+5,0) and (0,+9.4)  now translating them by R2 and D3 gives us new points at (2,5),(2-11),(-3,-3),(7,-3),(2,-12.4),(2,6.4) • Our equation becomes (x-2)2/25 - (y+3)2/64 = -1

  8. (D) In-Class Examples – The Graph

  9. (E) Internet Links • http://www.analyzemath.com/EquationHyperbola/EquationHyperbola.html - an interactive applet fom AnalyzeMath • http://home.alltel.net/okrebs/page63.html - Examples and explanations from OJK's Precalculus Study Page • http://tutorial.math.lamar.edu/AllBrowsers/1314/Hyperbolas.asp - Ellipses from Paul Dawkins at Lamar University • http://www.webmath.com/hyperbolas.html - Graphs of ellipses from WebMath.com

  10. (F) Homework • AW, p540, Q3abc, 5cd, 8, 17, 23 • Nelson text, p616, Q1-5eol,7,12,15,16

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