Homographic Functions

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Homographic Functions avril 09 纪光 - 北京景山学校 - Homographic Functions

纪光 - 北京景山学校 - Homographic Functions Basic type (Review 1) A > 0 when x  +∞ then y  0 (+) when x  -∞ then y  0 (-) x-axis y = 0 is an asymptote for (H) when x  0 (+) then y  +∞ when x  0 (-) then y  -∞ y-axis x = 0 is an asymptote for (H) The vertex of the Hyperbola is the point (√A,√A) on the Axis (y=x). The function is an odd function O is the center of symetry of (H). avril 09 纪光 - 北京景山学校 - Homographic Functions

纪光 - 北京景山学校 - Homographic Functions Basic type (Review 2) A < 0 when x  +∞ then y  0 (-) when x  -∞ then y  0 (+) x-axis y = 0 is an asymptote for (H) when x  0 (+) then y  - ∞ when x  0 (-) then y  + ∞ y-axis x = 0 is an asymptote for (H) The vertex of the Hyperbola is the point (-√(-A),√(-A) on the Axis (y=-x). The function is an odd function O is the center of symetry of (H). avril 09 纪光 - 北京景山学校 - Homographic Functions

First transformation (1) avril 09 纪光 - 北京景山学校 - Homographic Functions

First transformation (1) h = +2 avril 09 纪光 - 北京景山学校 - Homographic Functions

First transformation (2) avril 09 纪光 - 北京景山学校 - Homographic Functions

First transformation (2) h = +2 avril 09 纪光 - 北京景山学校 - Homographic Functions

纪光 - 北京景山学校 - Homographic Functions 2nd transformation (1) A > 0 A > 0 avril 09 纪光 - 北京景山学校 - Homographic Functions

纪光 - 北京景山学校 - Homographic Functions 2nd transformation (1) A > 0 avril 09 纪光 - 北京景山学校 - Homographic Functions

纪光 - 北京景山学校 - Homographic Functions 2nd transformation (2) A < 0 A < 0 avril 09 纪光 - 北京景山学校 - Homographic Functions

纪光 - 北京景山学校 - Homographic Functions 2nd transformation (2) A < 0 avril 09 纪光 - 北京景山学校 - Homographic Functions

纪光 - 北京景山学校 - Homographic Functions 3rd transformation A > 0 avril 09 纪光 - 北京景山学校 - Homographic Functions Homographic Functions

Change of center and variables Let X = x – l and Y = y – h then the equation becomes which means that, with respect to the new center 0’(l,h), the graph of the function is the same as the original. avril 09 纪光 - 北京景山学校 - Homographic Functions

纪光 - 北京景山学校 - Homographic Functions Limits & Asymptotes when x  +∞ or x  - ∞ then y  h (±) the line y = h is an asymptote for (H) when x  l (±) then y  ±∞ the line x = l is an asymptote for (H) The point (l,h) intersection of the two asymptotes is the center of symmetry of the hyperbola. avril 09 纪光 - 北京景山学校 - Homographic Functions

纪光 - 北京景山学校 - Homographic Functions General case It’s easy to check that all functions in the type of can be changed into the form of f5(x). Example : Problem : prove that all functions defined by : can be transformed into the previous one. Example : avril 09 纪光 - 北京景山学校 - Homographic Functions

纪光 - 北京景山学校 - Homographic Functions General case In this example l = 1, h = 4, A = 9 «Horizontal» Asymptote : y = 4 «Vertical» Asymptote : x = 1 Center : (1;4). A > 0  function is decreasing. Only one point is necessary to be able to place the whole graph ! Interception with the Y-Axis : (0,-5) or Interception with the X-Axis : avril 09 纪光 - 北京景山学校 - Homographic Functions

纪光 - 北京景山学校 - Homographic Functions General case Formulas : l = and h = In fact one can find the asymptotes by looking for the limits of the function in the original form. Then it’s not necessary to change the form to be able to plot the graph. avril 09 纪光 - 北京景山学校 - Homographic Functions

纪光 - 北京景山学校 - Homographic Functions 祝好运谢谢再见 avril 09 纪光 - 北京景山学校 - Homographic Functions