The standard equation of the parabola is:
[tex]y=\frac{x^2}{6}+x+3[/tex]Explanation:The focus of the parabola, (h, f) = (-3, 3)
The directrix: y = 0
The equation of a parabola is of the form:
[tex]y=\frac{1}{4(f-k)}(x-h)^2+k[/tex]The distance from the focus to the vertex is equal to the distance from the vertex to the directrix
f - k = k - y
3 - k = k - 0
k + k = 3 + 0
2k = 3
k = 3/2
Substitute k = 3/2, f = 3, and h = -3 into the equation above
[tex]\begin{gathered} y=\frac{1}{4(3-\frac{3}{2})}(x-(-3))^2+\frac{3}{2} \\ y=\frac{1}{4(\frac{3}{2})}(x+3)^2+\frac{3}{2} \\ y=\frac{1}{6}(x+3)^2+\frac{3}{2} \\ y=\frac{(x+3)^2}{6}+\frac{3}{2} \end{gathered}[/tex]This can be further simplified as:
[tex]\begin{gathered} y=\frac{x^2+6x+9}{6}+\frac{3}{2} \\ y=\frac{x^2+6x+9+9}{6} \\ y=\frac{x^2+6x+18}{6} \\ y=\frac{x^2}{6}+\frac{6x}{6}+\frac{18}{6} \\ y=\frac{x^2}{6}+x+3 \end{gathered}[/tex]Therefore, the standard equation of the parabola is:
[tex]y=\frac{x^2}{6}+x+3[/tex]