Współrzędne wierzchołka W = (p, q) paraboli będącej wykresem funkcji:
y = ax² + bx + c możemy obliczyć ze wzorów:
[tex]\bold{p=\dfrac{-b}{2a}\ ,\qquad q=\dfrac{-\Delta}{4a}}[/tex] , gdzie Δ = b² - 4ac
a)
[tex]\bold{ y=4x^2+6x+3\quad\implies\quad a=4\,,\quad b=6\,,\quad c=3}\\\\\Delta=6^2-4\cdot4\cdot3=36-48=-12\\\\\bold{p=\dfrac{-6}{2\cdot4}=-\dfrac34\ ,\qquad q=\dfrac{-(-12)}{4\cdot4}=\dfrac34}\\\\\\ \large\boxed{\bold{W=\left(-\frac34\,,\ \frac34\right)}}[/tex]
b)
[tex]\bold{ y=-2x^2+4x+1\quad\implies\quad a=-2\,,\quad b=4\,,\quad c=1}\\\\\Delta=4^2-4\cdot(-2)\cdot1=16+8=24\\\\\bold{p=\dfrac{-4}{2\cdot(-2)}=\dfrac44=1\ ,\qquad q=\dfrac{-24}{4\cdot(-2)}=\dfrac62=3}\\\\\\ \large\boxed{\bold{W=\left(1,\, \big 3\right)}}[/tex]
c)
[tex]\bold{ y=\frac12x^2-x+1\quad\implies\quad a=\frac12\,,\quad b=-1\,,\quad c=1}\\\\\Delta=(-1)^2-4\cdot\frac12\cdot1=1-2=-1\\\\\bold{p=\dfrac{-(-1)}{2\cdot\frac12}=\dfrac11=1\ ,\qquad q=\dfrac{-(-1)}{4\cdot\frac12}=\dfrac12}\\\\\\ \large\boxed{\bold{W=\left(1\,,\ \frac12\right)}}[/tex]
