Equation of parabola with focus (f₁, f₂) and directrix ax+by+c=0 is:
[tex]\dfrac{(ax+by+c)^2}{a^2+b^2}=(x-f_1)^2+(y-f_2)^2[/tex]
So we have:
focus: (-2,4) ⇒ f₁ = -2, f₂ = 4
directrix: y = 2 ⇒ y - 2 = 0 ⇒ 0x + 1y - 2 = 0 ⇒ a = 0, b = 1, c = -2
and equation:
[tex]\dfrac{(ax+by+c)^2}{a^2+b^2}=(x-f_1)^2+(y-f_2)^2\\\\\\
\dfrac{(0x+1y-2)^2}{0^2+1^2}=(x-(-2))^2+(y-4)^2\\\\\\
\dfrac{(y-2)^2}{1}=(x+2)^2+(y-4)^2\\\\\\(y-2)^2=(x+2)^2+(y-4)^2\\\\y^2-4y+4=(x^2+4x+4)+(y^2-8y+16)\\\\
y^2-4y+4=x^2+4x+4+y^2-8y+16\quad|-y^2-4\\\\
-4y=x^2+4x-8y+16\quad|+8y\\\\-4y+8y=x^2+4x+16\\\\4y=x^2+4x+16\quad|:4\\\\\boxed{y=\frac{1}{4}x^2+x+4}[/tex]
I don't know which equation of a parabola you know, so i used general one (the one that always works).
