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Funksies in die vorm y = a x + q

Funksies met die algemene vorm y = a x + q word reguitlyn funksies genoem. In die vergelyking, y = a x + q , is a en q konstantes en het verskillende invloede op die grafiek van die funksie. Die algemene grafiek van so 'n funksie word gegee in [link] vir die funksie f ( x ) = 2 x + 3 .

Grafiek van f ( x ) = 2 x + 3

Ondersoek: funksies van die vorm y = a x + q

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

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

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

Die verskillende eienskappe word opgesom in [link] .

Opsomming van algemene vorms en posisies van grafieke van funksies in die vorm y = a x + q
a > 0 a < 0
q > 0
q < 0

Definisieversameling en waardeversameling

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

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

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

Afsnitte

Vir funksies van die vorm, y = a x + q word die metode om die afsnitte met die x - en y -asse te bereken, uiteengesit.

Die y -afsnitte word as volg bereken:

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

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

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

Die x -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

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

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

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 ) = a x + q

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

  1. die teken van a
  2. y -afsnit
  3. x -afsnit

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

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

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

Die y -afsnit word bepaal deur x = 0 te stel en is vroeër bereken as y i n t = - 1 . Die x -afsnit word bepaal deur y = 0 te stel en is vroeër bereken as x i n t = 1 .

Grafiek van die funksie g ( x ) = x - 1

Teken die grafiek van y = 2 x + 2 .

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

    y = 2 ( 0 ) + 2 = 2
  2. Om die x-afsnit te kry, stel y = 0 .

    0 = 2 x + 2 2 x = - 2 x = - 1

Afsnitte

  1. Skryf die y -afsnitte neer vir die volgende reguitlyngrafieke:
    1. y = x
    2. y = x - 1
    3. y = 2 x - 1
    4. y + 1 = 2 x
  2. Gee die vergelyking van die grafiek wat hieronder geskets is:
  3. Skets die volgende verbande op dieselfde assestelsel, merk die koördinate van die afsnitte duidelik: x + 2 y - 5 = 0 en 3 x - y - 1 = 0

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Source:  OpenStax, Siyavula textbooks: wiskunde (graad 10) [caps]. OpenStax CNX. Aug 04, 2011 Download for free at http://cnx.org/content/col11328/1.4
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