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Funksies van die vorm y = a b ( x ) + q

Funksies van die vorm y = a b ( x ) + q is bekend as eksponensiële funksies. Die algemene vorm van ‘n funksie van hierdie tipe word gewys in [link] .

Algemene vorm en posisie van die grafiek van ‘n funksie met die vorm f ( x ) = a b ( x ) + q .

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

  1. Op dieselfde assestelsel, skets die volgende grafieke:
    1. a ( x ) = - 2 . b ( x ) + 1
    2. b ( x ) = - 1 . b ( x ) + 1
    3. c ( x ) = - 0 . b ( x ) + 1
    4. d ( x ) = - 1 . b ( x ) + 1
    5. e ( x ) = - 2 . b ( x ) + 1
    Gebruik jou antwoorde om 'n gevolgtrekking ten opsigte van die invloed van a te maak.
  2. Op dieselfde assestelsel, skets die volgende grafieke:
    1. f ( x ) = 1 . b ( x ) - 2
    2. g ( x ) = 1 . b ( x ) - 1
    3. h ( x ) = 1 . b ( x ) + 0
    4. j ( x ) = 1 . b ( x ) + 1
    5. k ( x ) = 1 . b ( x ) + 2
    Gebruik jou antwoorde om 'n gevolgtrekking ten opsigte van die invloed van q te maak.

Jy sou gevind het dat die waarde van a bepaal die vorm van die grafiek, dit wil sê: “Curves Upwards” – “CU” ( a > 0 ) of “Curves Downwards” – “CD” ( a < 0 ).

Jy sou ook gevind het die waarde van q bepaal die posisie van die y -afsnit.

Hierdie verskillende eienskappe word opgesom in [link] .

Getabelleerde opsomming van algemene vorms en posisies van funksies van die vorm y = a b ( x ) + q
a > 0 a < 0
q > 0
q < 0

Definisieversameling en waardeversameling

Vir y = a b ( x ) + q , is die funksie gedefinieer vir alle reële waardes van x . Dus, die definisieversameling is { x : x R } .

Die waardeversameling van y = a b ( x ) + q word bepaal deur die teken van a .

As a > 0 dan:

b ( x ) 0 a · b ( x ) 0 a · b ( x ) + q q f ( x ) q

Dus, as a > 0 , dan is die waardeversameling { f ( x ) : f ( x ) [ q ; ) } .

As a < 0 dan:

b ( x ) 0 a · b ( x ) 0 a · b ( x ) + q q f ( x ) q

Dus, as a < 0 , dan is die waardeversameling { f ( x ) : f ( x ) ( - ; q ] } .

Byvoorbeeld, die definisieversameling van g ( x ) = 3 . 2 x + 2 is { x : x R } . Vir die waardeversameling,

2 x 0 3 · 2 x 0 3 · 2 x + 2 2

Dus is die waardeversameling { g ( x ) : g ( x ) [ 2 ; ) } .

Afsnitte

Vir funksies van die vorm, y = a b ( x ) + q , word die afsnitte met die x en y -as bereken deur x = 0 te stel vir die y -afsnit en deur y = 0 te stel vir die x -afsnit.

Die y -afsnit word as volg bereken:

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

Byvoorbeeld, die y -afsnit van g ( x ) = 3 . 2 x + 2 word gegee deur x = 0 te stel, om dan te kry:

y = 3 . 2 x + 2 y i n t = 3 . 2 0 + 2 = 3 + 2 = 5

Die x -afsnitte word bereken deur y = 0 te stel, soos volg:

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

Dit het net ‘n rëele oplossing as een van beide a < 0 of q < 0 . Anders, het die grafiek van die vorm y = a b ( x ) + q geen x -afsnitte.

Byvoorbeeld, die x -afsnit van g ( x ) = 3 . 2 x + 2 word gegee deur y = 0 te stel:

y = 3 · 2 x + 2 0 = 3 · 2 x i n t + 2 - 2 = 3 · 2 x i n t 2 x i n t = - 2 3

en dit het geen rëele oplossing nie. Dus, die grafiek van g ( x ) = 3 . 2 x + 2 het geen x -afsnitte nie.

Asimptote

Daar is een asimptoot vir funksies van die vorm y = a b ( x ) + q . Die asimptoot kan bepaal word deur die analise van die waardeversameling.

Ons het gesien dat die terrein bepaal word deur die waarde van q. As a > 0 , dan is die terrein { f ( x ) : f ( x ) [ q ; ) } .

Dit wys dat die funksiewaarde neig na die waarde van q as x . Dus die horisontale asimptoot lê by y = q .

Sketse van grafieke van die vorm f ( x ) = a b ( x ) + q

Om grafieke te skets van funksies van die vorm, f ( x ) = a b ( x ) + q , moet ons vier eienskappe bereken:

  1. Definisieversameling en Waardeversameling
  2. y -afsnit
  3. x -afsnit

Byvoorbeeld, skets die grafiek van g ( x ) = 3 . 2 x + 2 . Merk die afsnitte.

Ons het die definisieversameling bepaal om { x : x R } te wees en die waardeversameling om { g ( x ) : g ( x ) ( 2 , ) } te wees.

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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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