$W=\left(p,q\right)\Rightarrow f\left(x\right)=a{\left(x-p\right)}^2+q$

 

$\text{a)}W\left(-1,4\right),P\left(0,3\right)$

$f\left(x\right)=a{\left(x+1\right)}^2+4$

$P=\left(0,3\right)\Rightarrow f\left(0\right)=3\Rightarrow a\cdot {\left(0+1\right)}^2+4=3\Rightarrow$  

$\Rightarrow a\cdot 1+4=3\Rightarrow a=3-4=-1$   

$\underline{\underline{f\left(x\right)=-{\left(x+1\right)}^2+4}}$ $\underline{\underline{f\left(x\right)=-{\left(x+1\right)}^2+4}}$

Przekształcamy wzór do postaci ogólnej:

$f\left(x\right)=-{\left(x+1\right)}^2+4=-\left(x^2+2x+1\right)+4=\underline{\underline{-x^2-2x+3}}$

 

$\text{b)}W\left(2,0\right),P\left(0,4\right)$

$f\left(x\right)=a{\left(x-2\right)}^2$

$P=\left(0,4\right)\Rightarrow f\left(0\right)=4\Rightarrow a\cdot {\left(0-2\right)}^2=4\Rightarrow 4a=4\Rightarrow a=1$

$\underline{\underline{f\left(x\right)={\left(x-2\right)}^2}}$

Przekształcamy wzór do postaci ogólnej:

$f\left(x\right)={\left(x-2\right)}^2=\underline{\underline{x^2-4x+4}}$

 

 

 

$\text{c)}W\left(4,-2\right),P\left(0,6\right)$

$f\left(x\right)=a{\left(x-4\right)}^2-2$

$P=\left(0,6\right)\Rightarrow f\left(0\right)=6\Rightarrow a\cdot {\left(0-4\right)}^2-2=6\Rightarrow a\cdot 16-2=6\Rightarrow a\cdot 16=8\Rightarrow a=\frac{8}{16}=\frac{1}{2}$

$\underline{\underline{f\left(x\right)=\frac{1}{2}{\left(x-4\right)}^2-2}}$

Przekształcamy wzór do postaci ogólnej:

$f\left(x\right)=\frac{1}{2}{\left(x-4\right)}^2-2=\frac{1}{2}\left(x^2-8x+16\right)-2=\frac{1}{2}{x}^2-4x+8-2=\underline{\underline{\frac{1}{2}{x}^2-4x+6}}$  

 

$\text{d)}W\left(2,7\right),P\left(0,-5\right)$

$f\left(x\right)=a{\left(x-2\right)}^2+7$

$P=\left(0,-5\right)\Rightarrow f\left(0\right)=-5\Rightarrow a\cdot {\left(0-2\right)}^2+7=-5\Rightarrow a\cdot 4+7=-5\Rightarrow a\cdot 4=-12\Rightarrow a=-3$

$\underline{\underline{f\left(x\right)=-3{\left(x-2\right)}^2+7}}$

Przekształcamy wzór do postaci ogólnej:

$f\left(x\right)=-3{\left(x-2\right)}^2+7=-3\left(x^2-4x+4\right)+7=-3x^2+12x-12+7=\underline{\underline{-3x^2+12x-5}}$