Answer:
The equation of the parabola is [tex]y=-\frac{x^2}{16}[/tex]
Step-by-step explanation:
We start by assuming a general point on the parabola [tex](x,y)[/tex].
Using the distance formula
[tex]\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex],
we find that the distance between [tex](x,y)[/tex] and the focus (0,-4) is
[tex]\sqrt{(x-0)^2+(y+4)^2}[/tex], and the distance between [tex](x,y)[/tex] and the directrix y =4 is [tex]\sqrt{(y-4)^2}[/tex]. On the parabola, these distances are equal:
[tex]\sqrt{(y-4)^2}=\sqrt{(x-0)^2+(y+4)^2}\\\\\mathrm{Square\:both\:sides}\\\\\left(\sqrt{\left(y-4\right)^2}\right)^2=\left(\sqrt{\left(x-0\right)^2+\left(y+4\right)^2}\right)^2\\\\(y-4)^2=(x-0)^2+\left(y+4\right)^2}\\\\y^2-8y+16=x^2+y^2+8y+16\\\\y^2-8y+16-16=x^2+y^2+8y+16-16\\\\y^2-8y=y^2+8y+x^2\\\\y^2-8y-\left(y^2+8y\right)=y^2+8y+x^2-\left(y^2+8y\right)\\\\-16y=x^2\\\\\frac{-16y}{-16}=\frac{x^2}{-16}\\\\y=-\frac{x^2}{16}[/tex]
