1. Homographic Functions 1avril 09 纪光 - 北京 景山学校 - Homographic Functions

2. Basic type (Review 1) 2avril 09 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).

3. Basic type (Review 2) 3avril 09 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).

4. First transformation (1) A = 1 4avril 09 纪光 - 北京 景山学校 - Homographic Functions A = 1

5. First transformation (1) A = 1 5avril 09 纪光 - 北京 景山学校 - Homographic Functions A = 1 h = +2

6. First transformation (2) A = -1 6 avril 09 纪光 - 北京 景山学校 - Homographic Functions A = -1

7. First transformation (2) A = -1 7 avril 09 纪光 - 北京 景山学校 - Homographic Functions A = -1 h = +2

8. 2 nd transformation (1) 8 avril 09 纪光 - 北京 景山学校 - Homographic Functions A > 0

9. 2 nd transformation (1) 9 avril 09 纪光 - 北京 景山学校 - Homographic Functions A > 0

10. 2 nd transformation (2) 10 avril 09 纪光 - 北京 景山学校 - Homographic Functions A < 0

11. 2 nd transformation (2) 11 avril 09 纪光 - 北京 景山学校 - Homographic Functions A < 0

12. 3 rd transformation 12 avril 09 纪光 - 北京 景山学校 - Homographic Functions A > 0

13. Change of center and variables 13 avril 09 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.

14. Limits & Asymptotes 14 avril 09 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.

15. General case 15 avril 09 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 :

16. General case 16 avril 09 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 :

17. General case 17 avril 09 纪光 - 北京 景山学校 - Homographic Functions 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.

18. 18 avril 09 纪光 - 北京 景山学校 - Homographic Functions 祝好运 谢谢 再见