1. The document describes methods for completing squares to graph functions and find their domains and ranges. 2. Examples are provided of using the method to graph several functions and find their corresponding domains and ranges. 3. For each function, the method involves completing the square of the variables, finding the x- and y-intercepts, and stating the domain and range.

1. FUNCIONES
MÉTODO DE COMPLETAR CUADRADOS
Graficar hallando dominio y rango de:
f = { ( x, y ) Î ¡ 2 / y = 3 x 2 + 6 x - 2}
y = 3x 2 + 6 x - 2
y + 2 = 3x 2 + 6 x
y + 2 = 3( x2 + 2 x )
y + 2 = 3 é( x + 1) - 1ù
2
ë û
y + 2 = 3 ( x + 1) - 3
2
y + 5 = 3 ( x + 1)
2
y = -5 Ù x = -1
v = ( -1, -5 )
Dom ( f ) = ﾡ
Ran ( f ) = ￩-5, ﾥ +
￫

2. 2. GRAFICAR HALLANDO DOMINIO Y RANGO DE:
F = {( x, y ) ￎ R 2 / X = -2 y 2 + 6 y - 9}
x = -2 y 2 + 6 y - 9
x + 9 = -2 y 2 + 6 y
x + 9 = -2( y 2 - 3 y )
3 9
x + 9 = [-2( y - ) 2 - ]
2 4
3 9
x + 9 = -2( y - ) 2 +
2 2
9 3
x + = -2( y - ) 2
2 2
9
x+ =0
2
9 3
x=- y- =0
2 2
3
y=
2
9 3
V = (- , )
2 2
9
Dom( f ) = ﾥ - , -
2
Ran( f ) = R

3. Graficar, hallando dominio y rango de
f = { ( x, y ) ￎ ﾡ 2 / 6 y 2 + 3 x + 12 y - 5 = 0}
3 x = -6 y 2 - 12 y + 5
3 x - 5 = -6 y 2 - 12 y
3 x - 5 = -6 ( y 2 + 2 y )
3 x - 5 = ￩-6 ( y + 1) - 1
2
￫
3 x - 5 = -6 ( y + 1) + 6
2
3 x - 11 = -6 ( y + 1)
2
￦ 11
= -6 ( y + 1)
2
3 x-
￨ 3
￦ 11
= -2 ( y + 1)
2
x-
￨ 3
11
x- =0
3 y +1 = 0
11 y = -1
x=
3
￦11
v= , -1
￨3
11
Dom ( f ) = -ﾥ ,
3
Ran ( f ) =ﾡ

4. Graficar, hallando dominio y rango de:
f = { ( x, y ) ￎ ﾡ 2 / 5 x 2 + 20 x + 6 y 2 + 36 y = 7}
5 x 2 + 20 x + 6 y 2 + 36 y = 7
5 ( x2 + 4x ) + 6 ( y 2 + 6 y ) = 7
5 ￩( x + 2 ) - 4 + 6 ￩( y + 3) - 9 = 7
2 2
￫ ￫
5 ( x + 2 ) - 20 + 6 ( y + 3) - 54 = 7
2 2
5 ( x + 2 ) + 6 ( y + 3) = 81
2 2
( x + 2) ( y + 3)
2 2
+ =1
81 81
5 6
2 2
￦ 81 ￦ 81
3, 7
￨ 5 ￨ 6
x+2=0 y+3= 0
x = -2 y = -3
v = ( -2, -3)

5. Graficar hallando dominio y rango de :
f = { ( x, y ) ￎ ﾡ 2 / 3x 2 + 12 x + 3 y 2 + 60 y = 7}
3 x 2 + 12 y + 3 y 2 + 60 y = 7
3 ( x 2 + 4 x ) + 3 ( y 2 + 20 y ) = 7
3 ￩( x + 2 ) - 4 + 3 ￩( y + 10 ) - 100 = 7
2 2
￫ ￫
3 ( x + 2 ) - 12 + 3( y + 10) 2 - 300 = 7
2
( x + 2 ) + 3 ( y + 10 ) = 7 + 300 + 12
2 2
3 ( x + 2 ) + 3 ( y + 10 ) = 319
2 2
319
( x + 2) + ( y + 10 ) =
2 2
3
x+2=0 y + 10 = 0
x = -2 y = -10

6. v = ( -2, -10 )
Dom ( f ) = ( -12,3,8,3)
Ran ( f ) = ( -20,3, 0,3)