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Funksies in die vorm y=ax+qy=ax+q

Funksies met die algemene vorm y=ax+qy=ax+q word reguitlynfunksies genoem. In die vergelyking, y=ax+qy=ax+q, is aa en qq konstantes en het verskillende invloede op die grafiek van die funksie. Die algemene grafiek van so 'n funksie word gegee in Figure 1 vir die funksie f(x)=2x+3f(x)=2x+3.

Figure 1: Grafiek van f(x)=2x+3f(x)=2x+3
Figure 1 (MG10C11_005.png)

Ondersoek: Funksies van die vorm y=ax+qy=ax+q

  1. Op dieselfde assestelsel, trek die volgende grafieke:
    1. a(x)=x-2a(x)=x-2
    2. b(x)=x-1b(x)=x-1
    3. c(x)=xc(x)=x
    4. d(x)=x+1d(x)=x+1
    5. e(x)=x+2e(x)=x+2
    Gebruik jou resultate om die invloed van verskillende waardes van qq op die resulterende grafiek af te lei.
  2. Op dieselfde assestelsel, teken die volgende grafieke:
    1. f(x)=-2.xf(x)=-2.x
    2. g(x)=-1.xg(x)=-1.x
    3. h(x)=0.xh(x)=0.x
    4. j(x)=1.xj(x)=1.x
    5. k(x)=2.xk(x)=2.x
    Gebruik jou resultate om die invloed van verskillende waardes van aa op die resulterende grafiek af te lei.

Jy behoort te vind dat die waarde van aa die helling van die grafiek beïnvloed. Soos aa vermeerder, vermeerder die helling van die grafiek ook. Indien a>0a>0 sal die grafiek vermeerder van links na regs (opwaartse helling). Indien a<0a<0 sal die grafiek verminder van links na regs (afwaartse helling). Dit is hoekom daar na aa verwys word as die helling of die gradiënt van 'n reguitlynfunksie.

Jy behoort ook te vind dat die waarde van qq die punt bepaal waar die grafiek die yy-as sny. Om hierdie rede, staan qq bekend as die y-afsnit.

Die verskillende eienskappe word opgesom in Table 1.

Table 1: Opsomming van algemene vorms en posisies van grafieke van funksies in die vorm y=ax+qy=ax+q
  a > 0 a > 0 a < 0 a < 0
q > 0 q > 0
Figure 2
Figure 2 (MG10C11_006.png)
Figure 3
Figure 3 (MG10C11_007.png)
q < 0 q < 0
Figure 4
Figure 4 (MG10C11_008.png)
Figure 5
Figure 5 (MG10C11_009.png)

Definisieversameling en Waardeversameling

Vir f(x)=ax+qf(x)=ax+q, is die definisieversameling {x:xR}{x:xR}, omdat daar geen waarde is van xRxR waarvoor f(x)f(x) ongedefinieërd is nie.

Die waardeversameling van f(x)=ax+qf(x)=ax+q is ook {f(x):f(x)R}{f(x):f(x)R} omdat daar geen waarde van f(x)Rf(x)R waarvoor f(x)f(x) ongedefinieërd is nie.

Byvoorbeeld, die definisieversameling van g(x)=x-1g(x)=x-1 is {x:xR}{x:xR} omdat daar geen waardes is van xRxR waarvoor g(x)g(x) ongedefinieërd is nie. Die waardeversameling van g(x)g(x) is {g(x):g(x)R}{g(x):g(x)R}.

Afsnitte

Vir funksies van die vorm, y=ax+qy=ax+q word die metode om die afsnitte met die xx- en yy-asse te bereken, uiteengesit.

Die yy-afsnitte word as volg bereken:

y = a x + q y i n t = a ( 0 ) + q = q y = a x + q y i n t = a ( 0 ) + q = q
(1)

Byvoorbeeld, die yy-afsnit van g(x)=x-1g(x)=x-1 word bepaal deur x=0x=0 te stel en dan op te los:

g ( x ) = x - 1 y i n t = 0 - 1 = - 1 g ( x ) = x - 1 y i n t = 0 - 1 = - 1
(2)

Die xx-afsnit word as volg bereken:

y = a x + q 0 = a · x i n t + q a · x i n t = - q x i n t = - q a y = a x + q 0 = a · x i n t + q a · x i n t = - q x i n t = - q a
(3)

Byvoorbeeld, die xx-afsnit van g(x)=x-1g(x)=x-1 word gegee deur y=0y=0 in te stel en dan op te los:

g ( x ) = x - 1 0 = x i n t - 1 x i n t = 1 g ( x ) = x - 1 0 = x i n t - 1 x i n t = 1
(4)

Draaipunte

Die grafiek van 'n reguitlynfunksie het nie draaipunte nie.

Simmetrie-asse

Die grafieke van reguitlynfunksies het gewoonlik nie simmerie-asse nie.

Skets van Grafieke van die vorm f(x)=ax+qf(x)=ax+q

Om die grafieke van die vorm f(x)=ax+qf(x)=ax+q te skets, moet ons die volgende drie eienskappe vind:

  1. die teken van aa
  2. yy-afsnit
  3. xx-afsnit

Slegs twee punte word benodig om 'n reguitlyn te trek. Die maklikste punte is die xx-afsnit (waar die lyn die xx-as sny) en die yy-afsnit.

Byvoorbeeld, skets die grafiek van g(x)=x-1g(x)=x-1. Merk duidelik die afsnitte.

Eerstens bereken ons dat a>0a>0. Dit beteken die grafiek gaan 'n opwaartse helling hê.

Die yy-afsnit word bepaal deur x=0x=0 te stel en is vroeër bereken as yint=-1yint=-1. Die xx-afsnit word bepaal deur y=0y=0 te stel en is vroeër bereken as xint=1xint=1.

Figure 6: Grafiek van die funksie g(x)=x-1g(x)=x-1
Figure 6 (MG10C11_010.png)

Exercise 1: Trek van 'n Reguitlyngrafiek

Teken die grafiek van y=2x+2y=2x+2.

Solution
  1. Step 1. Vind die y-afsnit :

    Om die y-afsnit te vind, stel x=0x=0.

    y = 2 ( 0 ) + 2 = 2 y = 2 ( 0 ) + 2 = 2
    (5)
  2. Step 2. Vind die x-afsnit :

    Om die x-afsnit te kry, stel y=0y=0.

    0 = 2 x + 2 2 x = - 2 x = - 1 0 = 2 x + 2 2 x = - 2 x = - 1
    (6)
  3. Step 3. Teken die grafiek deur die twee koördinate te vind en dan te verbind. :

    Figure 7
    Figure 7 (MG10C11_011.png)

Afsnitte

  1. Skryf die yy-afsnitte neer vir die volgende reguitlyngrafieke:
    1. y=xy=x
    2. y=x-1y=x-1
    3. y=2x-1y=2x-1
    4. y+1=2xy+1=2x
    Kliek hier vir die oplossing
  2. Gee die vergelyking van die grafiek wat hieronder geskets is:
    Figure 8
    Figure 8 (MG10C11_012.png)
    Kliek hier vir die oplossing
  3. Skets die volgende verbande op dieselfde assestelsel, merk die koördinate van die afsnitte duidelik: x+2y-5=0x+2y-5=0 en 3x-y-1=03x-y-1=0
    Kliek hier vir die oplossing

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