MATHHX B

MATHHX B

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4.7 Skæring mellem polynomier

I dette afsnit skal vi se på, hvordan man beregner skæringspunkter mellem polynomier. Vi starter dog lige med at repetere, hvordan man bestemmer skæringspunkter ved aflæsning.

  • Eksempel 4.7.1
    Lad \(f(x)=-x^2+2\) og lad \(g(x)=-2x+2\). Vi vil nu bestemme skæringspunktet. Vi tegner graferne og markerer de to skæringspunkter med grøn:

    (-tikz- diagram)

    Vi aflæser de to skæringspunkter til at være \((0,2)\) og \((2,-2)\).

Det er nemt at aflæse skæringspunkterne, men hvis gerne vil respekteres af sin matematiklærer, bør man også kunne beregne dem.

  • Eksempel 4.7.2
    Vi vil nu beregne skæringspunkterne for funktionerne fra eksempel 4.7.1. Det gøres ved at sætte funktionerne lig hinanden og løse den ligning der fremkommer. Vi skal altså løse ligningen:

    \[f(x)=g(x)\]

    Vi indsætter forskrifterne:

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

    og løser ligningen som en andengradsligning. Først skal vi skal omforme den, så der står nul på den ene side:

    \[-x^2+2x=0\]

    Vi finder vi diskriminanten:

    \begin{align*} d &= b^2-4ac \\ & =2^2-4\cdot (-1)\cdot 0 \\ & =4 \end{align*} Vi bestemmer så de to løsninger:

    \[x_1=\frac {-b-\sqrt {d}}{2a}=\frac {-2-\sqrt {4}}{2\cdot (-1)}=\frac {-4}{-2}=2\]

    og

    \[x_2=\frac {-b+\sqrt {d}}{2a}=\frac {-2+\sqrt {4}}{2\cdot (-1)}=\frac {0}{-2}=0\]

    Altså er løsningen til ligningen \(x_1=2\) og \(x_2=0\).

    Nu har vi fundet \(x\)-værdierne til de to skæringspunkter. Vi mangler så bare \(y\)-værdierne og dem finder vi ved at sætte \(x\)-værdierne ind i en af forskrifterne. Vi bestemmer selv, hvilken en forskrift vi bruger. Vi vælger at sætte dem ind i \(g(x)\). Vi får den første y-værdi til:

    \[g(0)=-2\cdot 0 +2=2,\]

    og den anden y-værdi til

    \[g(2)=-2\cdot 2+2=-2.\]

    Vi får altså skæringspunkterne til \((0,2)\) og \((2,-2)\), hvilket er det samme som vi fik ved aflæsning. Denne gang er vi så bare sikre på, at vi har de helt præcise punkter. Vi er hermed blevet til rigtige matematikere.

Øvelse 4.7.1

Bestem ved beregning skæringspunkterne mellem følgende funktioner:

  • a) \(f(x)=x^2\quad \) og \(\quad g(x)=4\)

  • b) \(f(x)=x^2+x-3\quad \) og \(\quad g(x)=x+1\)

  • c) \(f(x)=2x+1\quad \) og \(\quad g(x)=-x+7\)

  • d) \(f(x)=2\) og \(g(x)=x^2+3\)

  • e) \(f(x)=x^3+x^2\) og \(g(x)=2x\) (svær)

Løsning 4.7.1

  • a) \((-2,4)\) og \((2,4)\)

  • b) \((-2,-1)\) og \((2,3)\)

  • c) \((2,5)\)

  • d) Der er ikke nogen.

  • e) \((-2,-4)\), \((0,0)\) og \((1,2)\)