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

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


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


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


A = 1


hx

AyxfH

x

AyxfH nTranslatio

Vertical

:)(:)( 2211

A = 1


A = 1h = +2


hx

AyxfH

x

AyxfH nTranslatio

Vertical

:)(:)( 2211

A = -1


A = -1


hx

AyxfH

x

AyxfH nTranslatio

Vertical

:)(:)( 2211

A = -1


A = -1h = +2

2nd transformation (1)

lx

AyxfH

x

AyxfH nTranslatio

Horizontal

:)(:)( 3311


A > 0

y 1

x

A > 0

y 1

x


lx

AyxfH

x

AyxfH nTranslatio

Horizontal

:)(:)( 3311


A > 0

y 1

x

y 1

x 2


lx

AyxfH

x

AyxfH nTranslatio

Horizontal

:)(:)( 3311


A < 0

y 4x

A < 0

y 4x


lx

AyxfH

x

AyxfH nTranslatio

Horizontal

:)(:)( 3311


A < 0

y 4x

y 4x 2

3rd transformationh

lx

AyxfH

x

AyxfH nTranslatio

hlV

:)(:)( 44

);(

11


A > 0

y 1

x

y 1

x 21

Change of center and variables

hlx

AyxfH

x

AyxfH nTranslatio

hlV

:)(:)( 44

);(

11


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


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


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


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


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


祝好运谢谢再见

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

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