MATHHX A

MATHHX A

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{\morecmidrules }{}\) \(\newcommand {\specialrule }[3]{\hline }\) \(\newcommand {\addlinespace }[1][]{}\) \(\def \LWRsiunitxrangephrase { \protect \mbox {to (numerical range)} }\) \(\def \LWRsiunitxdecimal {.}\)

3.5 Parabler

I de følgende afsnit skal vi se nærmere på de geometriske figurer der dukker op i forbindelse med kvadratisk programmering. Vi starter med parablen, da I kender den allerede. Vi husker at en parabel er grafen for et andengradspolynomium

\[f(x)=ax^2+bx+c,\]

hvor \(a\neq 0\). Når man konstruerer en graf, bliver funktions værdier \(f(x)\) til \(y\)-værdier for grafen, og derfor kan ligningen for parablen skrives som

\[y=ax^2+bx+c\]

  • Eksempel 3.5.1
    Ligningen \(x^2+y=3\) er ligning for en parabel. Det kan vi se ved at omskrive den så den får den sædvanlige form. Det gør vi ved at isolere \(y\):

    \[y=-x^2+3\]

    Vi ser at den nu har form som en parabel \(y=ax^2+bx+c\).

Øvelse 3.5.1

Betragt ligningen \(2x^2-2y+x+4=0\)

  • a) Omskriv ligningen så den har form som en parabel.

Løsning 3.5.1

  • a) \(y=x^2+\frac {1}{2}x+2\)

Øvelse 3.5.2

Betragt ligningen \(2x=3y-1\)

  • a) Undersøg om ligningen er ligning for en parabel.

Løsning 3.5.2

  • a) Isolere vi \(y\) får vi \(y=\frac {2}{3}x+\frac {1}{3}\). Det er en lineær funktion, så det er ikke en parabel (husk at vi kræver at \(a\neq 0\) i parablens ligning)

Har vi en parabel og et punkt kan vi afgøre om punktet ligger på parablen.

  • Eksempel 3.5.2
    Vi vil nu undersøge om punktet \((1,3)\) ligger på parablen \(y=2x^2-x+2.\) Vi indsætter punktets koordinater i stedet for \(x\) og \(y\) i ligningen:

    \[3=2\cdot 1^2-1+2,\]

    og reducerer

    \[3=3.\]

    Da ligningen passer må punktet ligge på parablen.

Øvelse 3.5.3

  • a) Undersøg om punktet \((2,2)\) ligger på parablen \(y=x^2-1\)

Løsning 3.5.3

  • a) nope

Øvelse 3.5.4

I skal opfriske betydningen af koefficienterne \(a\), \(b\) og \(c\) i forskriften for andengradspolynomiet. Hvilken en betydning har de for grafen?

Løsning 3.5.4

Koefficienten \(a\) viser om parablen er konveks (glad) eller konkav (sur). Er \(a\) positiv er den konveks, ellers er den konkav. Derudover viser \(a\) hvor "spids"grafen er. Er \(a\) tæt på nul er parablen meget flad.

Koefficienten \(b\) har betydning for placeringen af grafen. Helt konkret viser den tangentens hældning i skæringspunktet med y-aksen.

Koefficienten \(c\) er angiver det sted hvor grafen skærer y-aksen. Men se kommentarer efter øvelsen.

Det er betydningen af \(a\) og især \(c\) som vi for brug for at kende til snart. I forhold til \(a\) er det vigtigt at kunne bestemme krumningen ud fra \(a\) (konveks eller konkav). I forhold til \(c\) plejer man at sige at grafen skærer grafen y-aksen i \(c\). Det er også rigtig nok, men vi har brug for at tænke på \(c\) på en lidt anden måde. Lad os sammenligne de to grafer \(y=2x^2-6x+5\) og \(y=2x^2-6x+7\). Vi kan se at de er ens bortset fra \(c\)-værdien. Men hvad betyder det i forhold til deres form og placering i koordinatsystemet? For en givet \(x\)-værdi er \(y\)-værdien for den anden graf 2 højere end \(y\)-værdierne for den første. Altså er de to grafer ens, bortset fra at den anden graf er forskudt med to opad. Så ændrer man på \(c\)-værdien i en parabel vil det forskyde grafen op eller ned alt efter om \(c\) bliver højere eller lavere.

Øvelse 3.5.5

  • a) Beskriv forskelle i form og placering af de to parabler \(y=-x^2+2x+1\) og \(y=-x^2+2x-4\).

Løsning 3.5.5

  • a) De har samme form, men den anden er forskudt med \(5\) nedad i forhold til den første.