Given the following question:
focus = (-2, 4)
directrix = y = 2
[tex]\begin{gathered} \frac{(ax+by+c)^2}{a^2+b^2}=(x-f1)^2+(y-f2)^2 \\ (-2,4)=(f1,f2) \\ y=2=y-2=0=0x+1y-2=0,a=0,b=1,c=-2 \\ \frac{(ax+by+c)^2}{a^2+b^2}=(x-f1)^2+(y-f2)^2 \\ \frac{0x+1y-2)^2}{0^2+1^2}=(x-(-2))^2_{}+(y-4)^2 \\ \frac{(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 \\ -4y=x^2+4x-8y+16+8 \\ -4y+8y=x^2+4x+16 \\ 4y=x^2+4x+16\colon4 \\ 4y=x^2+4x+16\colon4=y=\frac{1}{4}x^2+x+4 \\ y=\frac{1}{4}x^2+x+4 \end{gathered}[/tex]
Now for the sketch:
