To solve the problem, it is best that we plot the given so we get an idea how the parabola looks.
The vertex of a parabola is between the directrix and the focus. Also, because we have a horizontal directrix, we know that the parabola opens downward.
Again, the vertex is right in between the directrix and the focus. So we know that the value of p is 1.5.
Using the values that we have so far, we can complete the equation. We have:
h = -2, k = 1.5, p = 1.5 direction of the parabola: opens downward
So the equation should be:
[tex]\begin{gathered} (x-h)^2=-4p(y-k) \\ \\ (x+2)^2=-4(1.5)(y-1.5) \\ \\ (x+2)^2=-6(y-1.5) \end{gathered}[/tex]
We rewrite this in the completing the square format using properties of equations.
[tex]\begin{gathered} (x+2)^{2}=-6(y-1.5) \\ \\ (x+2)^2=-6y+9 \\ \\ 6y=-(x+2)^2+9 \\ \\ y=-\frac{1}{6}(x+2)^2+\frac{3}{2} \end{gathered}[/tex]
