Một số phương pháp vẽ đồ thị của hàm số có chứa dấu giá trị tuyệt đối

Một số phương pháp vẽ đồ thị của hàm số có chứa dấu giá trị tuyệt đối... PHƯƠNG PHÁP KHẢO SÁT HAØM SỐ CÓ CHỨA DẤU GIÁ TRỊ TUYỆT ĐỐI Dạng 1.[r]

! "#$ %

D ng 1 a vào th hàm s ( ) : C y = f x ( ) &'( th hàm s

( C ) : y = f x ( )

0 ( ) :

0

≥

* -*+ -* /01 ( C1) : y1 = f x ( ) )*+ 2 0 -* /01 ,

3 0 , 4 5 0 -* /01 ( ) : C y = f x ( ) 67 08 / 9

3 0 2, 4 5 0 -* /01 ( ) : C y = f x ( ) 67 08 :;<+ 9

4 =( -*= 9;+ >' 9

D ng 2 a vào th hàm s ( ) : C y = f x ( ) &'( th hàm s

( C ) : y = f x ( )

0 9 +/ , ( C2) : y2 = f x ( ) 4 5 0 57 &*= )0 ?

( C ) : y = f x ( ) 0 ( 4 57 / '@) -*= 9;+ A

f x y

f x

* -*+ -* /01 ( C2) : y2 = f x ( ) )*+ 2 0 -* /01 ,

3 0 , 4 5 0 -* /01 ( ) : C y = f x ( ) 67 08 B 0 C (

3 0 2, 4 5 0 -* /01 4 =( -*= 9;+ >' ( DE 0 57 &*= )0 ?

D ng 3 a vào th hàm s ( ) : C y = f x ( ) &'( th hàm s

( C ) : y = f x ( )

0 9 +/ , =' M x y ( ;0 0) ∈ ( C3) M x ( ;0 − y0) ∈ ( C3)

( C ) : y = f x ( ) 0 9 4 57 / '@) -*= 9;+ A )*+, ( C3) : y3 = f x ( ) = y y3 = y y ≥ 0

* -*+ -* /01 ( C3) : y3 = f x ( ) )*+ 2 0 -* /01 ,

3 0 , 4 5 0 -* /01 ( ) : C y = f x ( ) 67 08 / 9

3 0 2, 4 5 0 -* /01 4 =( -*= 9;+ >' 9 A

D ng 4 a vào th hàm s ( ) : C y = f x ( ) = u x v x ( ) ( ) &'(

th hàm s ( C4) : y4 = u x v x ( ) ( )

)*+,

4 4

* -*+ -* /01 ( C4) : y4 = u x v x ( ) ( ) )*+ 2 0 -* /01 ,

3 0 , 4 5 0 -* /01 ( ) : C y = f x ( ) 67 / 7 u x ( ) ≥ 0

3 0 2, 4 5 0 -* /01 ( ) : C y = f x ( ) 67 / 7 u x ( ) ≤ 0

4 =( -*= 9;+ >' 9

a vào th hàm s ( ) : C y = f x ( ) = ( x − a v x ) ( )

&'( th hàm s ( C4) : y4 = x − a v x a ( ), ∈

)*+,

4 4

* -*+ -* /01 ( C4) : y4 = x − a v x a ( ), ∈

)*+ 2 0 -* /01 ,

3 0 ,

4 5 0 -* /01 ( ) : C y = f x ( ) 67 B 0 C -;<5 /0 F 9 G

3 0 2, 4 5 0 -* /01 ( ) : C y = f x ( ) 67 B / +

-;<5 /0 F 9 G 4 =( -*= 9;+ >' 9A

;5 I : @ -* /01 )*+ )0;+ : =' + / 1 /'( / -*= )< B C / / )*+ /0 J &'(

0 ' : @ -* /01 )*+ )0;+ : =' + / 1 /'( / -*= K0 +) )0 F 0 @ ,

D ng 5 a vào th hàm s ( ) : C y = f x ( ) &'( th hàm s

( C ) : y = f x ( )

% J D L ( C5) : y5 = f x ( ) / 4 57 2 B;<+) 0; & ',

3 M;<+) , D L y51 = f x ( ) = g x ( ) :;@ D 5* : @ 2

3 M;<+) 2, D L y5 = f x ( ) = g x ( ) :;@ D 5* : @

D ng 6 a vào th hàm s ( ) : C y = f x ( ) &'( th hàm s

( C ) : y = f x ( )

% J D L ( C6) : y6 = f x ( ) / 4 57 2 B;<+) 0; & ',

3 M;<+) , D L y61 = f x ( ) = g x ( ) :;@ D 5* : @ 2

3 M;<+) 2, D L y6 = g x ( ) :;@ D 5* : @ H

D ng 7 a vào th hàm s ( ) : C y = f x ( ) &'( th hàm s

( C ) : y = f x ( )

% J D L ( C7) : y7 = f x ( ) / 4 57 H B;<+) 0; & ',

3 M;<+) , D L y71 = f x ( ) = g x ( ) :;@ D 5* : @ 2

3 M;<+) 2, D L y72 = f x ( ) = g x ( ) = h x ( ) :;@ D 5* : @

3 M;<+) H, D L ( C7) : y7 = h x ( ) :;@ D 5* : @ H

0* 0 57 &*= 3 2

y = x − x + )*+ -* /01 Q RA

0 C* & +/ D 5 D L -* /01 Q R )'C 0 57 &*=A

N =/ 0;< / E 0 / = /'( = )'C -* /01 Q R / @ * - J7 )'C Q R D<+ -;<5 /0 F 9 G − A

HR E7 /0 7 &*= 7 - J 0;< / E 0 3 2

2 x − 3 x + 2 = m )*+ B*=

C

0 C* & +/ D 5 D L -* /01 Q R )'C 0 57 &*=A

2

' 6 6

y = x − x y'=0⇔ x=0 x = 1

! " # $ %$ # $& $& x=0;y =1 ! " # $ %$ & ' $& x=1;y =0

limx y

→±∞

= ±∞

T 3 S U S 3

+∞

−∞

'' 12 6

y = x− y'' = 0 ⇔ x= ()

Q W2X W2R

N =/ )'C -* /01 Q R / @ * - J7 )'C Q R D<+ -;<5 /0 F 9 G −

9 G − GY ( G ZQ− R G −I GY * - J7 Q − X−IR

/// )*+ : @ , y = f ' ( x0).( x − x0) + y0A

0

'( ) '( 1) 12

f x = f − = GY /// ,y = 12( x + 1) − 4 = 12 x + 8A

-5 -4 -3 -2 -1 1 2 3 4 5

-5 -4 -3 -2 -1

1 2 3 4 5

x y

P

Q

O

* − − + , )

y = x − x +

/ 0

#& '! 1 2 !

! # & %3

4

E7 /0 7 &*= 7 - J 0;< / E 0 3 2

2 x − 3 x + 2 = m )*+ B*= 0 7

0 B /A

2 x − 3 x + 2 = m ⇔ 2 x − 3 x + 1 = m − 1

% ( 4 5 % % )'C -* /01 (C1), 3 2

y = x − x + D 5 -;<5 /0 F :, ( G 7−

)*+(C1),

3 2

y

=

GY (C1) )*+ 2 0 -* /01,

0 , %* /01 Q R 67 B 0 C / '@) ( Q) C - J7 67 / (R

0 , V =( -*= 9;+ -* /01 0 >' (

DE 0 57 &*= y1 4 5 0 57 &*= )0 ?

N L (C1)Q E 0 2R

-5 -4 -3 -2 -1 1 2 3 4 5

-5 -4 -3 -2 -1

1 2 3 4 5

x

y

Q

O

3 2

4 )

;@ D 5* (C1) / )*+, S \ 7 − \ \GY \ 7 \ 2

0* 0 57 &*= 1 4 2

2

y= x − x + )*+ -* /01 4 5

0 C* & +/ D 5 D L -* /01 )'C 0 57 &*=A

%1 0 - J 0;< / E 0 , 1 4 2

B /A

%1 0 - J 0;< / E 0 , 1 4 2

2

B /A

C

!" # $ % &

_%, 57 &*= )0 ?

3

' 2 8

<+ 0 @ ,xlim→±∞ y

= +∞

MM ,

" "

2

y = x − X y'' = 0 ⇔x= ± 2 3 / 3

M_

$ % & $ '( & $ %

3 / 3# ! ) 3 / 3# ! )

%* /01,

o *+, -* /01 0 , 4 57 / '@) -*= 9;+

o %%M, - !# " / !# "

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

x

y

' ()

← →

1

4 3 2

y= x − x +

← →

← →

" M \GY− 5 < lgm< 3 \GY 5 3 5 3

lg10− < lgm< lg10 ⇔ 10− <m< 10

)R %1 0 - J 0;< / E 0 , 1 4 2

2

% ( 4 5 % % )'C -* /01 (C1), 4 2

1

1

2

:, ( G 7−

)*+ , 1 1

0 ( ) :

0

≥

* -*+ -* /01 ( C1) : y1 = f x ( ) )*+ 2 0 -* /01 ,

3 0 , 4 5 0 -* /01 ( ) : C y = f x ( ) 67 08 / 9

3 0 2, 4 5 0 -* /01 ( ) : C y = f x ( ) 67 08 :;<+ 9

4 =( -*= 9;+ >' 9

-5 -4 -3 -2 -1

1 2 3 4 5

x y

1

1

2

" M \GY0 < lgm< 3 \GY 3

lg1 lg < m< lg10 ⇔ 1 <m< 1000

1 N L th hàm s

2

( ) :

1

=

−

x

x

D L -* /01 0 57 &*=

2

( ) :

1

=

−

x

x

-5 -4 -3 -2 -1 1 2 3 4 5

-3 -2 -1 1 2 3 4 5 6 7 8

x

y

2

( ) :

1

=

−

x

C y

x

;@ D 5* Q R / )*+,

2

1 1

( ) :

1

=

−

x

x )*+ 2 0 -* /01 ,

3 0 , 4 5 0 -* /01 ( ) : C y = f x ( ) 67 B 0 C -;<5 /0 F 9 G

3 0 2, 4 5 0 -* /01 ( ) : C y = f x ( ) 67 B / +

-;<5 /0 F 9 G 4 =( -*= 9;+ >' 9A

-3 -2 -1

1 2 3 4 5 6 7 8

x

y

2

( ) :

1

=

−

x

x

1 ( ) :

1

−

= +

x

x

D L -* /01 0 57 &*= ( ) : 1 1

−

= +

x

x

-5 -4 -3 -2 -1

1 2 3 4 5

x

y

1 ( ) :

1

−

= +

x

x

1 ( ) :

1

−

= +

x

x )*+ 2 0 -* /01 ,

3 0 , 4 5 0 -* /01 ( ) : C y = f x ( ) 67 08 / 9

3 0 2, 4 5 0 -* /01 4 =( -*= 9;+ >' 9 A

-5 -4 -3 -2 -1 1 2 3 4 5

-5 -4 -3 -2 -1

1 2 3 4 5

x

y

1 1

1 ( ) :

1

−

= +

x

x

2 N L th hàm s

2

5 5

( ) :

1

=

−

x

x

;@ D 5* -* /01 0 57 &*=

2

( ) :

1

=

−

x

x <C D8 :'@ H / )*+,

2

5 5

( ) :

1

=

−

x

x )*+ 2 0 -* /01 ,

3 0 , 4 5 0 -* /01 ( ) : C y = f x ( ) 67 08 B 0 C (

3 0 2, 4 5 0 -* /01 4 =( -*= 9;+ >' ( DE 0 57 &*= )0 ?

-5 -4 -3 -2 -1

1 2 3 4 5 6 7 8

x

y

2

( ) :

1

=

−

x

x

3 N L th hàm s

2

( ) :

1

=

−

x

x

;@ D 5* -* /01 0 57 &*=

2

5 5

( ) :

1

=

−

x

x <C D8 :'@ [ / )*+,

( ) :

1

=

−

x

x )*+ 2 0 -* /01 ,

3 0 , 4 5 0 -* /01 ( C5) 67 08 / 9

3 0 2, 4 5 0 -* /01 ( C5) 67 08 :;<+ 9

4 =( -*= 9;+ >' 9

-5 -4 -3 -2 -1 1 2 3 4 5

-5 -4 -3 -2 -1

1 2 3 4 5 6 7 8

x

y

2

( ) :

1

=

−

x

x

4 N L th hàm s

2

( ) :

1

=

−

x

x

;@ D 5* -* /01 0 57 &*=

2

( ) :

1

=

−

x

x <C D8 :'@ ] / )*+,

2

( ) :

1

=

−

x

x )*+ 2 0 -* /01 ,

3 0 , 4 5 0 -* /01 ( C6) 67 08 / 9

3 0 2, 4 5 0 -* /01 4 =( -*= 9;+ >' 9 A

-7 -6 -5 -4 -3 -2 -1

1 2 3 4 5 6 7

x

y

2

1

=

−

x