homographic functions 1avril 09 纪光 - 北京 景山学校 - homographic functions
TRANSCRIPT
Homographic Functions
x
AyxfH :)( 11
hx
AyxfH :)( 22
lx
AyxfH
:)( 33
hlx
AyxfH
:)( 44
dcx
baxyxfH
:)( 55
1avril 09 纪光 - 北京 景山学校 - Homographic Functions
Basic type (Review 1)
x
AyxfH :)( 11
2avril 09 纪光 - 北京 景山学校 - Homographic Functions
A > 0
y 1
x
• 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).
Basic type (Review 2)
x
AyxfH :)( 11
3avril 09 纪光 - 北京 景山学校 - Homographic Functions
A < 0
y 4x
• 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 (1)
hx
AyxfH
x
AyxfH nTranslatio
Vertical
:)(:)( 2211
A = 1
4avril 09 纪光 - 北京 景山学校 - Homographic Functions
A = 1
First transformation (1)
hx
AyxfH
x
AyxfH nTranslatio
Vertical
:)(:)( 2211
A = 1
5avril 09 纪光 - 北京 景山学校 - Homographic Functions
A = 1h = +2
First transformation (2)
hx
AyxfH
x
AyxfH nTranslatio
Vertical
:)(:)( 2211
A = -1
6avril 09 纪光 - 北京 景山学校 - Homographic Functions
A = -1
First transformation (2)
hx
AyxfH
x
AyxfH nTranslatio
Vertical
:)(:)( 2211
A = -1
7avril 09 纪光 - 北京 景山学校 - Homographic Functions
A = -1h = +2
2nd transformation (1)
lx
AyxfH
x
AyxfH nTranslatio
Horizontal
:)(:)( 3311
8avril 09 纪光 - 北京 景山学校 - Homographic Functions
A > 0
y 1
x
A > 0
y 1
x
2nd transformation (1)
lx
AyxfH
x
AyxfH nTranslatio
Horizontal
:)(:)( 3311
9avril 09 纪光 - 北京 景山学校 - Homographic Functions
A > 0
y 1
x
y 1
x 2
2nd transformation (2)
lx
AyxfH
x
AyxfH nTranslatio
Horizontal
:)(:)( 3311
10avril 09 纪光 - 北京 景山学校 - Homographic Functions
A < 0
y 4x
A < 0
y 4x
2nd transformation (2)
lx
AyxfH
x
AyxfH nTranslatio
Horizontal
:)(:)( 3311
11avril 09 纪光 - 北京 景山学校 - Homographic Functions
A < 0
y 4x
y 4x 2
3rd transformationh
lx
AyxfH
x
AyxfH nTranslatio
hlV
:)(:)( 44
);(
11
12avril 09 纪光 - 北京 景山学校 - Homographic Functions
A > 0
y 1
x
y 1
x 21
Change of center and variables
hlx
AyxfH
x
AyxfH nTranslatio
hlV
:)(:)( 44
);(
11
13avril 09 纪光 - 北京 景山学校 - Homographic Functions
y 1
x 21
Let X = x – l and Y = y – hthen 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.
Y AX
Limits & Asymptotes
(H4 ) y A
x l h
14avril 09 纪光 - 北京 景山学校 - Homographic Functions
y 1
x 21
• 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
dcx
baxyxfH
:)( 55
15avril 09 纪光 - 北京 景山学校 - Homographic Functions
y A
x l h
• Problem : prove that all functions defined by :can be transformed into the previous one.
y ax bcx d
Example :
y 1
x 21
x 1x 2
Example :
y 4x 5x 1
4(x 1) 9x 1
9
x 1 4
General case
dcx
baxyxfH
:)( 55
16avril 09 纪光 - 北京 景山学校 - Homographic Functions
y 4x 5x 1
4(x 1) 9x 1
9
x 1 4
• 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 :
( 54 ;0)
General case
dcx
baxyxfH
:)( 55
17avril 09 纪光 - 北京 景山学校 - Homographic Functions
hlx
A
dcx
baxy
• 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.
c
d
c
a
18avril 09 纪光 - 北京 景山学校 - Homographic Functions
祝好运谢谢再见