MATHHX B
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4.2 Parabler
Vi har set at der findes polynomier af alle mulige grader. Nulte og førstegradspolynomier er jo bare lineære funktioner, og dem ved vi allerede en del om, så næste skridt er at lære om andengradspolynomier. I dette afsnit skal vi se
nærmere på grafen for et andengradspolynomium. Et andengradspolynomium er en funktion på formen
\[f(x)=ax^2+bx+c,\]
hvor \(a \neq 0\) (betyder at \(a\) ikke må være nul).
Grafen for et andengradspolynomium hedder en parabel . Parablens udsende og placering afhænger af koefficienterne \(a\), \(b\) og \(c\). Koefficienten \(a\) bestemmer grafens form og \(b\)
og \(c\) har betydning for grafens placering.
Betydning af \(a\)
Parablen kan se ud på to forskellige måder alt efter værdien af koefficienten \(a\). Hvis \(a>0\) ligner parablen en glad mund og polynomiet siges at være konveks . Hvis \(a<0\) ligner parablen en sur mund og
polynomiet kaldes konkav :
For \(a>0\) er funktionen konveks
For \(a<0\) er funktionen konkav
Det kan være svært at huske hvad der er konveks og konkav, så man kan alternativt sige glad og sur (sorry Jytte). Ellers er der en huskeregel: Vi kan se, at et konvekst andengradspolynomium har en stigende
vækst og derfor er KonVÆKST. Det konkave polynomium ligner en som har slået sig, og er derfor KonkAV!
Øvelse 4.2.1
Afgør om følgende funktioner er konvekse eller konkave. Kan du gøre det både ved at tegne i GeoGebra og ved at betragte forskriften?
a) \(f(x)=x^2-3x+1\)
b) \(f(x)=-3x^2+4\)
Ud over at afgøre om polynomiet er konvekst eller konkavt fortæller \(a\) også hvor ”spids” parablen er. Her ses tre polynomier som er ens bortset fra størrelse \(a\)-værdien:
\(a=\frac {1}{2}\)
\(a=1\)
\(a=4\)
Der gælder tilsvarende for negative \(a\)-værdier:
\(a=-\frac {1}{2}\)
\(a=-1\)
\(a=-4\)
Så jo længere væk \(a\) ligger fra nul, jo spidsere bliver parablen.
Betydning af \(c\)
Koefficienten \(c\) er der hvor parablen skærer \(y\)-aksen.
Betydning af \(b\)
Koefficienten \(b\) er den sværeste. For at forklare betydningen skal vi først tegne en tangent til parablen. En tangent er en linje, som ligger op ad en graf i et punkt. Vi skal tegne tangenten gennem det punkt, hvor grafen skærer
\(y\)-aksen. Det ser således ud:
Vi kan se at tangenten (den røde linje) følger parablen omkring parablens skæringspunkt med \(y\)-aksen. Koefficienten \(b\) er hældningen på denne tangent.
Eksempel 4.2.1
Betragt grafen for et andengradspolynomium \(f\):
Vi ser at polynomiet er konkavt, så \(a\) må være negativ.
Parablen skærer \(y\)-aksen i \(1\), så \(c=1\).
Vi tegner nu en tangent til \(f\) gennem skæringspunktet med \(y\)-aksen for at finde \(b\):
Vi aflæser hældningen på tangenten:
Vi ser at hældningen på tangenten er \(2\), så \(b=2\).
Konklusion: \(a<0\), \(c=1\) og \(b=2\).
Øvelse 4.2.2
Betragt grafen for et polynomium \(f\):
Bestem fortegn for \(a\), samt værdien af \(b\) og \(c\).
Løsning 4.2.2
a) \(a>0\)
b) \(c=2\)
c) \(b=-1\)
Ekstra
Vi har set, hvordan man kan bestemme \(b\) og \(c\) ud fra grafen, men ikke hvordan man bestemmer \(a\). Dvs. vi har lært at bestemme fortegnet for \(a\), men ikke selve værdien. Det er dog ikke så svært. Vi finder \(a\) ved
at aflæse, hvor meget grafen vokser, når vi går en ud ad \(x\)-aksen med udgangspunkt i toppunktet.
Eksempel 4.2.2
Betragt graferne for to andengradspolynomier \(f\) og \(g\):
Hvis vi starter i toppunket og går \(1\) til højre ser vi at første graf falder med \(2\) og anden graf stiger med \(1\):
Vi konkludere at \(a=-2\) for det først polynomium og \(a=1\) for det andet polynomium.
Øvelse 4.2.3
Betragt graferne for to polynomier \(f\) og \(g\):
Løsning 4.2.3
a) \(a=\frac {1}{2}\)
b) \(a=-4\)
Vi har nu lært hvordan man aflæser, \(a\), \(b\) og \(c\) ud fra grafen, så vi er nu i stand til at bestemme hele forskriften ud fra grafen.
Øvelse 4.2.4
Betragt graferne for to polynomier \(f\) og \(g\):
Løsning 4.2.4
a) \(f(x)=-x^2+2x+1\)
b) \(g(x)=3x^2\)