Postać ogólna: [latex]f(x)=ax^{2}+bx+c[/latex] Postać iloczynowa: [latex]f(x)=a(x-x_{1})(x-x_{2})[/latex] Postać kanoniczna (wierzchołkowa): [latex]f(x)=a(x-p)^{2}+q[/latex] ===================================================== [latex]a) f(x)=2x^{2}+12x+10[/latex] ---> Postać kanoniczna: [latex]p=frac{-b}{2a}=frac{-12}{4}=-3\ \ q=frac{-Delta}{4a}=frac{-64}{8}=-8\ \ Delta=b^{2}-4a\ Delta=12^{2}-4*2*10\ Delta=144-80\ Delta=64\ \ f(x)=2(x+3)^{2}-8[/latex] ---> Postać iloczynowa: [latex]Delta=64\ sqrt{Delta}=8\ \ x_{1}=frac{-b-sqrt{Delta}}{2a}=frac{-12-8}{4}=frac{-20}{4}=-5\ \ x_{2}=frac{-b+sqrt{Delta}}{2a}=frac{-12+8}{4}=frac{-4}{4}=-1\ \ f(x)=2(x+5)(x+1)[/latex] ===================================================== [latex]b) f(x)=-frac{1}{2}(x-6)^{2}-1[/latex] ---> Postać ogólna: [latex]f(x)=-frac{1}{2}(x-6)^{2}-1\ \ f(x)=-frac{1}{2}(x^{2}-12x+36)-1\ \ f(x)=-frac{1}{2}x^{2}+6x-18-1\ \ f(x)=-frac{1}{2}x^{2}+6x-19[/latex] ---> Postać iloczynowa: [latex]Delta=6^{2}-4*(-frac{1}{2})*(-19)\ Delta=36-38\ Delta==-2[/latex] Δ=-2<0 - brak pierwiastków; funkcji nie można zapisać w postaci iloczynowej ===================================================== [latex]c) f(x)=-3(x-2)(x+1)[/latex] ---> Postać ogólna: [latex]f(x)=-3(x-2)(x+1)\ \ f(x)=-3(x^{2}-x-2)\ \ f(x)=-3x^{2}+3x+6[/latex] ---> Postać kanoniczna: [latex]p=frac{-3}{-6}=frac{1}{2}\ \ q=frac{-81}{-3*4}=frac{27}{4}\ \ Delta=3^{2}-4*(-3)*6\ Delta=9+72\ Delta=81\ \ f(x)=-3(x-frac{1}{2})^{2}+frac{27}{4}[/latex] ===================================================== [latex]d) f(x)=5x^{2}-15x[/latex] ---> Postać kanoniczna: [latex] p=frac{15}{10}=-frac{3}{2}\ \ q=frac{-225}{5*4}=-frac{45}{4}\ \ Delta=(-15)^{2}-4*5*0\ Delta=225\ \ f(x)=5(x+frac{3}{2})^{2}-frac{45}{4}[/latex] ---> Postać iloczynowa: [latex]f(x)=5x^{2}-15x\ \ f(x)=5x(x-3)[/latex]
