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En funktion er en sammenhæng mellem to variable størrelser, x og y. En funktion beskriver hvordan en afhængig variable størrelse, som også kaldes y, varierer som en konsekvens af ændringer i en anden, såkaldt uafhængig variable størrelse, som også betegnes som x.

Sammenhængen mellem disse to variable kan betegnes med regneforskriften y = f(x).

Regneforskriften kan forklares således at der til en bestemt værdi af x kun én værdi af y.

b) Forklar, hvad de to tal a og b står for i den lineære funktion, og vis med grafiske eksempler, hvordan forskellige værdier af de to tal giver forskellige placeringer af funktionens graf.

A står for hældningskoefficienten det vil sige hvordan den lineære funktion hælder

B står for skæring i y asken.

F(x) = 2x+ 7

G(x)= 4x – 9

Se bilag

c) Forklar, hvordan man ud fra to kendte punkter på en graf for en lineær funktion kan bestemme en forskrift for funktionen. Det er vigtigt at du skriver formler og viser hvordan de kan bruges.

Forskriften for den lineære funktion y = ax + b, hvis graf går gennem punkterne (x1y1) og (x2y2), kan bestemmes ud fra følgene.

Hældningskoefficienten: a = y2-y1x2-x1

Skæring med y-aksen: b = y1 – a * x1

Eks. På bestemmelse af forskriften ud fra to punkter på grafen:

Den lineære graf går gennem punkterne (2,2) og (4,6). Vi kan beregne forskriften vha. overstående formler. Idet vi sætter x1 = 2; y1 = 2; x2 = 4; y2=6

A = 6-24-2= 2

B = 2-2*2=-2

Den lineære forskrift (y=ax+b) er derfor y = 2x – 2

d) Forklar om reglerne for at løse ligninger og uligheder.

Regler for løsninger af ligninger: 1. Man må reducere venstre og højre side hver for sig 2. Man må lægge samme tal til eller trække samme tal fra på begge sider af lighedstegnet 3. Man må gange/dividere med samme tal (dog ikke 0) på begge sider

Regler for løsning af uligheder:

Ovenstående regler gælder også for uligheder.

Samtidig gælder følgende regler: 1. Man må reducere venstre og højre side hver for sig 2. Man må lægge samme tal til eller trække samme tal fra på begge sider af ulighedstegnet, eller: man må flytte et led fra ene side af et ulighedstegn til den anden side, hvis man samtidig skifter fortegn 3. Man må gange eller dividere med samme positive tal på begge sider af et ulighedstegn 4. Man må gange eller dividere med negative tal på begge sider af et ulighedstegn, hvis man samtidig vender ulighedstegnet.

Opgaver 1. Løs følgende ligning: 2x-2=5-x

2x – 4 = 5 – x

2x = 9 – x

3x = 9

X = 93 2. Løs følgende ligning: 6x-2x-2-1=2x+126+4x+1

6x -2x + 4 – 1 = 2x + 3 + 2x + 1

4x + 3 = 4x + 4

4x = 4x + 1

4x – 4x = 1

X = 1

3. Løs følgende ulighed: 23x+2>-2(x-3)+4 23x + 2 > -2x + 6 + 4

23x > -2x + 8

23x + 2x > 8

223 x > 8

4. Bestem ved beregning løsning til de to ligninger med to ubekendte: 2x+3y=8 og 2x-4y=-6.

2x + 3y = 8

3y = 8 – 2x

Y = 83 + -2x3

2x – 4y = - 6

-4y = - 6 -2x

Y = -6-4 + -2x-4

83 + -2x3 = -6-4 + -2x-4

8*43*4 + -2x*43*4 = -6*-3-4*-3 + -2x*-3-4*-3

32 – 8x= 18 + 6x

-8x= -14 + 6x

-14x = -14

X = 1

Y = -6-4 + -2*1-4

Y = 1,5 + 0,5

Y = 2

5. Bestem ved beregningsmetoden forskriften for den lineære funktion, der er fastlagt ved følgende funktionsværdier f(-3) = -3 og f(6) = 0.

X1= - 3

Y1= - 3

X2= 6

Y2= 0

A = y2-y1x2-x1

A = 0+36+3

A = 39

A = 0,33

B = y1 – a * x1

B = -3 – 0,33*-3

B= -0,33

Y = ax + b

Y = 0,33x – 0,33

6. Bestem en forskrift for den lineære funktion, hvis graf går igennem punkterne (-4,-1) og (8, -4).

Se bilag

X1 = - 4

Y1 = -1

X2 = 8

Y2 = - 4

A = y2-y1x2-x1

A = -4-18-4

A = -54

A = -1,25

B = y1 – a * x1

B = -1 + 1,25 * -4

B = -6

Y = ax + b

Y = -1,25x - 6

7. Bestem skæringspunktet mellem graferne for funktionerne fx=-12x-2 og gx=x+1.

Se bilag.

Skæringspunktet = (-2,-1)

Anvendelse

Opgave 1

Lad prisen pr. kg for en bestemt vare være en funktion af afsætningen i kg. Afsætningen er givet ved forskriften fx=200-12x Antag endvidere, at prisen pr. kg som funktion af udbuddet i kg er givet ved gx=50+2x a) Tegn graferne for f og g i samme koordinatsystem.

(Se bilag, Lavet i geogebra) b) Bestem ligevægtsprisen, dvs. den pris, hvor udbud og afsætning er lige store. (altså skæringspunktet)

Skæringspunkt = (60,170)

Ligevægtsprisen er derfor 170 kr.

Opgave 2

Omkostningerne ved produktion af en pakke med 10 stk. vaffelis består dels af en fast omkostning på 10.000 kr. og dels en variabel omkostning på 25,50 kr. pr. pakke. Virksomheden sælger en pakke vaffelis til en fast pris på 38 kr.

Lad x betegne antallet af pakker med 10 stk. vaffelis. Lad f(x) være de samlede omkostninger og lad g(x) være den samlede omsætning i kr.

a) Bestem forskrifterne for f og g.

F(X) = 25,5x + 10.000

G(X) = 38x

b) Hvor mange pakker vaffelis skal virksomheden sælge for ikke at få underskud.

c) Tegn graferne for f og g i samme koordinatsystem og vis hvordan du kan aflæse svaret på spørgsmål b i koordinatsystemet.…...

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