homographic functions 1review sept.2010 纪光 - 北京 景山学校 - homographic functions
TRANSCRIPT
Homographic Functions
x
AyxfH :)( 11
(H2 ) f2 :x a y=
Ax
+ H
(H 3) f3 : x a y=
Ax−L
(H 4 ) f4 : x a y=
Ax−L
+ H
dcx
baxyxfH
:)( 55
1Review sept.2010 纪光 - 北京 景山学校 - Homographic Functions
Basic type (Review 1)
x
AyxfH :)( 11
2Review sept.2010 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 blue Axis (y=x).• The function is an odd function• O is the center of symetry of (H).
Basic type (Review 2)
x
AyxfH :)( 11
3Review sept.2010 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 (p.1)
(H1) f1: x a y =
Ax
VerticalTranslation⏐ →⏐ ⏐ ⏐ (H2 ) f2 : xa y=
Ax
+ H
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 = 1H = +2
(H1) f1: x a y =
Ax
VerticalTranslation⏐ →⏐ ⏐ ⏐ (H2 ) f2 : xa y=
Ax
+ H
First transformation (p.2)
(H1) f1:x a y =
Ax
VerticalTranslation⏐ →⏐ ⏐ ⏐ (H2 ) f2 :xa y=
Ax
+ H
A = -1
6Review sept.2010 纪光 - 北京 景山学校 - Homographic Functions
A = -1
First transformation (p.3)
(H1) f1:x a y =
Ax
VerticalTranslation⏐ →⏐ ⏐ ⏐ (H2 ) f2 :xa y=
Ax
+ H
A = -1
7Review sept.2010 纪光 - 北京 景山学校 - Homographic Functions
A = -1h = +2
2nd transformation (p.1)
8Review sept.2010 纪光 - 北京 景山学校 - Homographic Functions
A > 0
y 1
x
A > 0
y 1
x
(H1) f1: x a y =
Ax
HorizontalTranslation⏐ →⏐ ⏐ ⏐ (H3) f3: xa y=
Ax−L
2nd transformation (p.1b)
9Review sept.2010 纪光 - 北京 景山学校 - Homographic Functions
A > 0
y 1
x
y 1
x 2
(H1) f1: x a y =
Ax
HorizontalTranslation⏐ →⏐ ⏐ ⏐ (H3) f3: xa y=
Ax−L
2nd transformation (p.2)
10Review sept.2010 纪光 - 北京 景山学校 - Homographic Functions
A < 0
y 4x
A < 0
y 4x
(H1) f1: x a y =
Ax
HorizontalTranslation⏐ →⏐ ⏐ ⏐ (H3) f3: xa y=
Ax−L
2nd transformation (p.2b)
11Review sept.2010 纪光 - 北京 景山学校 - Homographic Functions
A < 0
y 4x
y 4x 2
(H1) f1: x a y =
Ax
HorizontalTranslation⏐ →⏐ ⏐ ⏐ (H3) f3: xa y=
Ax−L
3rd transformation
(H1) f1: x a y =
Ax
Vur(L;H )
Translation⏐ →⏐ ⏐ ⏐ (H4 ) f4 : xa y=A
x−L+ H
12Review sept.2010 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
13Review sept.2010 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
14Review sept.2010 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
15Review sept.2010 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
16Review sept.2010 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
17Review sept.2010 纪光 - 北京 景山学校 - Homographic Functions
y=ax+bcx+d
=A
x−L+ H
• In fact one can find the asymptotes by looking for the limits of the function in the original form.
L =−dc
H =ac
Then it’s not necessary to change the equation to be able to plot the graph.