CHAPTER 2 GRAPH FUNCTIONS.doc

8/19/2019 CHAPTER 2 GRAPH FUNCTIONS.doc

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CHAPTER 2 : GRAPHS OF FUNCTIONS II

Important Notes:

Kertas 1 (Obje t!"e #$est!ons%& soa'an b!asa ber !sar tt) bent$ * e+$+$ an b) )ra, $tama ba)!-INEAR& #UA.RATIC& CU/IC an+ RECIPROCA- FUNCTIONS0

Pelajar perlu menguasi utk i+ent!, :a% t e s ape o, )rap )!"en a t pe o, ,$n t!onb% t e t pe o, ,$n t!on )!"en a )rap

% t e )rap )!"en a ,$n t!on an+ "! e "ersa0

JENIS GRAF FUNGSI for Paper 1, SPM Mat emati!s"

Linear# $ m% & !

m $ gra'ient! $ #(inter!ept

Quadratic# $ a%) & *% & !

Note : Paper 1 limits to y = ax 2 + c

Cubic# $ a%+ &*%) &!% & '

Note : Paper 1 limits to y = ax 3 + c

Reciprocal

# $a

x

Note " is mo'ule !on!entrates on grap sket! ing, t e skills nee'e' for Paper 1- For 'etails on hoto plot !raphs please refer to t e ot er relate' mo'ules .eg- / apter 1) 0 e Straig t ine 0 on o2 to'ra2 3 plot straig t lines4

/a to /as!

I- -!near Grap s " !an *e represente' *# t e e5uation 3 m4 5 -• m is t e gra'ient• ! t e #(inter!ept, ie, t e pla!e 2 ere t e straig t line !uts t e #(a%is-

II- e general form of a 6ua'rati! Fun!tion is "#x$ = ax 2 + bx + c 7 a, *, ! are !onstants an' a % &- / ara!teristi!s of a 5ua'rati! fun!tion"

• In8ol8es one 8aria*le onl#,• e ig est po2er of t e 8aria*le is 20

III- e general form of a /u*i! Fun!tion is "#x$ = ax 3 + bx 2 + cx + d 7 a, *, ! an' ' are !onstants an'a 9 :-

/ ara!teristi!s of a !u*i! fun!tion"• In8ol8es one 8aria*le onl#,• e ig est po2er of t e 8aria*le is +-

I;- e simple Re!ipro!al Fun!tion is of t e form # $a

x, 2 ere a is a !onstant-

a x

O x

y

x

x

y

O

y

O x

y

O x



201 #$a+rat! Grap

E4er !se 1 :

a4 /omplete t e follo2ing ta*le for t e e5uation <=) ) +−= x x y

*4 ># using a s!ale of ) !m to 1 unit on t e %(a%is an' )!m to < unit on t e #(a%is, 'ra2 t e grapof <=) ) +−= x x y for ?: ≤≤ x

!4 From #our grap , fin'i4 t e 8alue of #, 2 en % $ 1-?ii4 t e 8alue of %, 2 en # $ 1<

'4 @ra2 a suita*le straig t line on #our grap to fin' t e 8alues of % 2 i! satisf# t e e5uation:A1:) ) =+− x x for ?: ≤≤ x - State t e 8alues of %-

E4er !se 2 :

a4 /omplete t e follo2ing ta*le for t e e5uation ( ) =<) −−= x x y

*4 ># using a s!ale of ) !m to 1 unit on t e %(a%is an' )!m to < unit on t e #(a%is, 'ra2 t e grapof ( ) =<) −−= x x y for <+ ≤≤− x

!4 @ra2 a suita*le straig t line on #our grap to fin' t e 8alues of % 2 i! satisf# t e e5uation( ) x x x )A=<) −=−− for <+ ≤≤− x - State t e 8alues of %-

E4er !se 6 :

a4 /omplete t e follo2ing ta*le for t e e5uation <)+ ) ++−= x x y

*4 ># using a s!ale of ) !m to 1 unit on t e %(a%is an' )!m to < unit on t e #(a%is, 'ra2 t e grapof <)+ ) ++−= x x y for A+ ≤≤− x

!4 From #our grap , fin'i4 t e 8alue of #, 2 en % $ (:-<ii4 t e 8alue of %, t at satisf# t e e5uation of <)+ ) =− x x

'4 @ra2 a suita*le straig t line on #our grap to fin' t e 8alues of % 2 i! satisf# t e e5uation:)<)+ )

=−+ x x for A+ ≤≤− x - State t e 8alues of %-

202 : C$b! Grap

x : 1 ) + < B ?

y < () (< 1 1: :

x (+ () (1 : 1 ) + <

y ) () (= (1) (11 + B

x (+ () (1 : 1 ) +

y 11 : (1B (+<



E4er !se 1:

a4 /omplete t e follo2ing ta*le for t e e5uation <1:+ +−= x x y

*4 ># using a s!ale of ) !m to 1 unit on t e %(a%is an' )!m to < unit on t e #(a%is, 'ra2 t e grapof <1:+ +−= x x y for <-+<-+ ≤≤− x

!4 From #our grap , fin' t e 8alue of #, 2 en % $ ()-<

'4 @ra2 a suita*le straig t line on #our grap to fin' t e 8alues of % 2 i! satisf# t e e5uationB1:+ =− x x for <-+<-+ ≤≤− x - State t e 8alues of %-

E4er !se 2:

a4 /omplete t e follo2ing ta*le for t e e5uation B1)+ +−= x x y

*4 ># using a s!ale of ) !m to 1 unit on t e %(a%is an' )!m to 1: unit on t e #(a%is, 'ra2 t e grapof B1)+ +−= x x y for A< ≤≤− x

!4 From #our grap , fin'i4 t e 8alue of #, 2 en % $ ()-<ii4 t e 8alues of positi8e %, 2 en # $ :

'4 @ra2 a suita*le straig t line on #our grap to fin' t e 8alues of % 2 i! satisf# t e e5uation

1)1)+

=− x x forAA ≤≤−

x - State t e 8alues of %-

E4er !se 6:

a4 /omplete t e follo2ing ta*le for t e e5uation 1)<+ −−= x x y

*4 ># using a s!ale of ) !m to 1 unit on t e %(a%is an' )!m to < unit on t e #(a%is, 'ra2 t e grap

of 1)<+

−−= x x y for <-++ ≤≤− x!4 From #our grap , fin'

i4 t e 8alue of #, 2 en % $ :-Cii4 t e 8alue of % 2 i! satisfies t e e5uation 1)<+ =− x x

'4 @ra2 a suita*le straig t line on #our grap to fin' t e 8alues of % 2 i! satisf# t e e5uationA=+ =− x x for <-++ ≤≤− x - State t e 8alues of %-

206 : Re !pro a' Grap

E4er !se 1:

x (+-< (+ () (1 : 1 ) + +-<

y ()-= C 1 < (? 1)-=

x (< ( (+ () (1 : 1 ) +

y (<= (1: )) 1? B (< (+ ))

x (+ () (1-< (1 : 1 ) + +-<

y (1: (?-= (C (1) (1B : 1+-



a4 /omplete t e follo2ing ta*le for t e e5uation x

yB

−=

*4 ># using a s!ale of ) !m to 1 unit on t e %(a%is an' )!m to ) unit on t e #(a%is, 'ra2 t e grap of

x y

B−= for AA ≤≤− x

!4 From #our grap ,i4 fin' t e 8alue of #, 2 en % $ 1-+ii4 fin' t e 8alue of %, 2 en # $ +-<


x x

=+ 1+

for AA ≤≤− x - State t e 8alues of %-

E4er !se 2:


y)

=

*4 ># using a s!ale of ) !m to 1 unit on t e %(a%is an' )!m to 1 unit on t e #(a%is, 'ra2 t e grap of

x y

)= for AA ≤≤− x

!4 From #our grap ,i4 fin' t e 8alue of #, 2 en % $ 1-<ii4 fin' t e 8alue of %, 2 en # $ (1-C


x x

A1) =+ for AA ≤≤− x - State t e 8alues of %-

E4er !se 6:

x ( ()-< (1 (:-B :-B 1 ) +

y 1-< B 1: (1: (B (+ (1-<

x ( (+ () (1 (:-< :-< 1 ) +

y (:-< (:-? (1 () 1 :-? :-<




yA

=

*4 ># using a s!ale of ) !m to 1 unit on t e %(a%is an' )!m to )unit on t e #(a%is, 'ra2 t e grap of

x y

A= for << ≤≤− x

!4 From #our grap ,i4 fin' t e 8alue of #, 2 en % $ (1-ii4 fin' t e 8alue of %, 2 en # $ -


:)

+=− x

x for A<-: ≤≤ x - State t e 8alues of %-

E4er !se 7:

a4 /omplete t e follo2ing ta*le for t e e5uation 1)

++=

x y

*4

>#using a s!ale of ) !m to :-< unit on t e %(a%is an' )!m to :-<unit on t e #(a%is, 'ra2 t e grap of

1)

++=

x y for A<-: ≤≤ x

!4 From #our grap ,

i4 fin' t e 8alue of #, 2 en % $ 1-+ii4 fin' t e 8alue of %, 2 en # $ )-C


:)

+=− x

x for A<-: ≤≤ x - State t e 8alues of %-

x (< ( () (1 (:-< :-< :-C 1-< )-< <

y (1 ( (C C < 1-B

x :-< 1 1-< ) )-< + +-<

y )-< 1-?< 1-B 1- + 1-+C

CHAPTER 2 GRAPH FUNCTIONS.doc

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