what is the equation of the parabola whose focus is (-3,1) and directrix is Y=4?

Because the directrix is a horizontal line, the parabola will be vertical.

Because the focus is below the directrix, the parabola will open downward.

The vertex of the parabola will be half-way between the focus and the directrix:  vertex  =  (-3, 2.5)

The focal length is the distance from the focus to the vertex:  focal length is:  f = 1.5

 

An equation for a parabola is:  y - y₁ =  a(x - x₁)  

     with a being found be the formula:  a  =  1/(4f)     --->     a  =  1/(4·1.5)     --->     a  =  1/6

     and (x₁, y₁) being the vertex     --->     x₁  =  -3     and     y₁  =  2.5.

 

Therefore, an equation is:  y - 2.5  =  (1/6)(x + 3)