Symmetrilinjen til andregradsfunksjoner er gitt ved:
[tex]Y=ax^2+bx+c[/tex]
[tex]y-c=a\left ( x^2+\frac{b}{a}x \right )[/tex]
[tex]y-c+a\left ( \frac{b^2}{4a^2} \right )=a\left ( x^2+\frac{b}{a}x+\frac{b^2}{4a^2} \right )[/tex]
[tex]y-c+\frac{b^2}{4a}=a\left ( x^2+\frac{b}{a}x+\frac{b^2}{4a^2} \right )[/tex]
[tex]y-\frac{4ac}{4a}+\frac{b^2}{4a}=a\left ( x+\frac{b}{2a} \right )^2[/tex]
[tex]y=a\left ( x-\left (- \frac{b}{2a} \right ) \right )^2+\left ( \frac{4ac-b^2}{4a} \right )\Rightarrow \left (-\frac{b}{2a},\frac{4ac-b^2}{4a} \right )[/tex]
Lektyre:
https://www.mathsisfun.com/algebra/comp ... quare.html
Alternativ:
[tex]f(x)=ax^2+bx+c[/tex]
[tex]f'(x)=2ax+b[/tex]
[tex]f'(x)=0\Longleftrightarrow 2ax+b=0\Longleftrightarrow x=-\frac{b}{2a}[/tex]
[tex]Ekstremalpunkt\, = \left ( -\frac{b}{2a},f\left ( -\frac{b}{2a} \right ) \right )[/tex]