Homographic Functions 1Review sept.2010 纪光 - 北京景山学校 - Homographic Functions.

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Homographic Functions 1Review sept.2010 纪光 - 北京景山学校 - Homographic Functions

Basic type (Review 1) 2Review sept.2010 纪光 - 北京景山学校 - Homographic Functions 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 blue Axis (y=x). The function is an odd function O is the center of symetry of (H).

Basic type (Review 2) 3Review sept.2010 纪光 - 北京景山学校 - Homographic Functions 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).

First transformation (p.1) A = 1 4Review sept.2010 纪光 - 北京景山学校 - Homographic Functions A = 1

First transformation (p.1b) A = 1 5Review sept.2010 纪光 - 北京景山学校 - Homographic Functions A = 1 H = +2

First transformation (p.2) A = -1 6 Review sept.2010 纪光 - 北京景山学校 - Homographic Functions A = -1

First transformation (p.3) A = -1 7 Review sept.2010 纪光 - 北京景山学校 - Homographic Functions A = -1 h = +2

2 nd transformation (p.1) 8 Review sept.2010 纪光 - 北京景山学校 - Homographic Functions A > 0

2 nd transformation (p.1b) 9 Review sept.2010 纪光 - 北京景山学校 - Homographic Functions A > 0

2 nd transformation (p.2) 10 Review sept.2010 纪光 - 北京景山学校 - Homographic Functions A < 0

2 nd transformation (p.2b) 11 Review sept.2010 纪光 - 北京景山学校 - Homographic Functions A < 0

3 rd transformation 12 Review sept.2010 纪光 - 北京景山学校 - Homographic Functions A > 0

Change of center and variables 13 Review sept.2010 纪光 - 北京景山学校 - Homographic Functions 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.

Limits & Asymptotes 14 Review sept.2010 纪光 - 北京景山学校 - Homographic Functions 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.

General case 15 Review sept.2010 纪光 - 北京景山学校 - Homographic Functions It’s easy to check that all functions in the type of can be changed into the form of f 5 (x). Problem : prove that all functions defined by : can be transformed into the previous one. Example :

General case 16 Review sept.2010 纪光 - 北京景山学校 - Homographic Functions 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 :

General case 17 Review sept.2010 纪光 - 北京景山学校 - Homographic Functions 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 equation to be able to plot the graph.