Phương pháp khảo sát hàm số chứa dấu giá trị tuyệt đối hiệu quả













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







 !"#$%

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Dng 1

a vào  th hàm s

( ) : ( )=C y f x

&'( th hàm s

1 1

( ) : ( )=C y f x



)*+,

1 1

( ) : 0

≥



= = − ≤



y y

C y y y y







*-*+-*./01

1 1

( ) : ( )=C y 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

2 2

( ) : ( )=C y f x



09+/,

2 2

( ) : ( )=C y f x

45057&*=)0?



2 2

( ) : ( )=C y f x

0(457/'@)-*=9;+A

)*+,

2 2

( ) 0 (1)

( ) : ( ) ( ) 0

= ≥



= = − ≤



f x y

C y f x f x

 





*-*+-*./01 2 2

( ) : ( )=C y 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

3 3

( ) : ( )=C y f x



09+/,='

0 0 3 0 0 3

( ; ) ( ) ( ; ) ( )∈− ∈

M x y C M x y C





3 3

( ) : ( )=C y f x

09457/'@)-*=9;+A

)*+, 3 3 3

( ) : ( ) 0= = = ≥C y f x y y y y



Traàn Phuù Vöông













2





*-*+-*./01

3 3

( ) : ( )=C y 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

4 4

( ) : ( ) . ( )=C y u x v x



)*+,

4 4

( ). ( ) ( ) ( ) 0

( ) : ( ) . ( ) ( ). ( ) ( ) ( ) 0

= = ≥



= = − = − = − ≤



u x v x f x y u x

C y u x v x u x v x f x y u x

 



*-*+-*./01

4 4

( ) : ( ) . ( )=C y u x v x

)*+20.-*./01,

30.,450.-*./01

( ) : ( )=C y f x

67/7.

( ) 0≥u x



30.2,450.-*./01

( ) : ( )=C y f x

67/7.

( ) 0≤u x



4=(-*=9;+>'9







a vào  th hàm s

( ) : ( ) ( ). ( )= = −C y f x x a v x



&'( th hàm s

4 4

( ) : . ( ),= − ∈ C y x a v x a



)*+,

4 4

( ). ( ) ( )

( ) : . ( ) ( ). ( ) ( )

− = = ≥



= − = − − = − = − ≤



x a v x f x y x a

C y x a v x x a v x f x y x a

 





*-*+-*./01

4 4

( ) : . ( ),= − ∈ C y 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



Traàn Phuù Vöông













H







;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

5 5

( ) : ( )=C y f x



%JDL

5 5

( ) : ( )=C y f x

/4572B;<+)0;&',

3M;<+),DL

( ) ( )= =y f x g x

:;@D5*:@2

3M;<+)2,DL

( ) ( )= =y f x g x

:;@D5*:@



Dng 6

a vào  th hàm s

( ) : ( )=C y f x

&'( th hàm s

6 6

( ) : ( )=C y f x



%JDL 6 6

( ) : ( )=C y f x

/4572B;<+)0;&',

3M;<+),DL

( ) ( )= =y f x g x

:;@D5*:@2

3M;<+)2,DL

( )=y g x

:;@D5*:@H



Dng 7

a vào  th hàm s

( ) : ( )=C y f x

&'( th hàm s

7 7

( ) : ( )=C y f x



%JDL

7 7

( ) : ( )=C y f x

/457HB;<+)0;&',

3M;<+),DL 71

( ) ( )= =y f x g x

:;@D5*:@2

3M;<+)2,DL

( ) ( ) ( )= = =y f x g x h x

:;@D5*:@

3M;<+)H,DL 7 7

( ) : ( )=C y h x

:;@D5*:@H



Traàn Phuù Vöông













I





$NOPP







0*057&*=

3 2

2 3 1y 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

2 3 2x x m− + =

)*+B*=

070B/A



C

0C*&+/D5DL-*./01QR)'C057&*=A





 







' 6 6y x x= −



' 0 0y x= ⇔ =



1x=







 −∞ 



+∞



 !"#$ %$#$&$&

0; 1x y= =



 !"#$%$& '$&

1; 0x y= =











lim

→±∞

= ±∞







 



−∞

S

+∞





3SUS3



+∞







−∞











'' 12 6y x= −



'' 0y x= ⇔ =

()



−∞



+∞



TUS3

V*.%V*L7

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

///)*+:@

,000

)).((' yxxxfy +−=

A

'( ) '( 1) 12f x f= − =

GY///

,

12( 1) 4 12 8y x x= + − = +

A



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

-5

-4

-3

-2

-1



* −−+

,)-

3 2

2 3 1y x x= − +

./0

#&'!12 !

!#&%3

4



Traàn Phuù Vöông













[





E7/07&*=7-J0;</E0 32

2 3 2x x m− + =

)*+B*=07

0B/A

)*+, 3 3

2 2

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



%(45%%)'C-*./01

( )C

,32

2 3 1y x x= − +

D5-;<5/0F

:,(G7

−



)*+

( )C

3 2

13 2

2 3 1 0

x x x

yx x x

− + ≥



=− − + <





 





GY

( )C

)*+20.-*./01,

0.,%*./01QR67B0C/'@)(Q)C-J767/(R

0.,V=(-*=9;+-*./010.>'(

DE057&*=

45057&*=)0?

NL

( )C

E02R

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

-5

-4

-3

-2

-1

2 3 1y x x= − +

4)





;@D5*

( )C

/)*+,S\7

−

\\GY\7\2 





0*057&*=

4 2

14 3

y x x= − +

)*+-*./0145





0C*&+/D5DL-*./01)'C057&*=A

Traàn Phuù Vöông

Phương pháp khảo sát hàm số có chứa dấu giá trị tuyệt đối

Tài liệu ôn tập môn toán tham khảo gồm một số phương pháp vẽ đồ thị của hàm số có chứa dấu giá trị tuyệt đối: Phương pháp khảo sát hàm số có chứa dấu giá trị tuyệt đối.

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