Answer

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**Hint:**In this question, we are going to find the vertex, focus and directrix of the parabola for the given equation.

The given equation is of the form of a parabola and now we are going to compare the given values to the standard form of a parabola.

By comparing those we get the value of vertex, focus and directrix of the parabola.

Hence, we can get the required result.

**Formula used:**If the parabola has a horizontal axis, the standard form of the equation of the parabola is

${(y - k)^2} = 4p(x - h)$, where $p \ne 0$

The vertex of this parabola is at $\left( {h,k} \right)$.

The focus is at $\left( {h + p,k} \right)$

The directrix is the line $x = h - p$

**Complete Step by Step Solution:**

In this question, we are going to find the vertex, focus and directrix for the given parabolic equation.

First write the given equation and mark it as $\left( 1 \right)$

$ \Rightarrow {\left( {x + \dfrac{1}{2}} \right)^2} = 4\left( {y - 3} \right)...\left( 1 \right)$

The given equation is of the parabolic form

We compare this equation to the standard form of the parabolic equation.

Here $p = 2$

The vertex V of the parabola is $\left( {\dfrac{{ - 1}}{2},3} \right)$

The focus of the parabola is $\left( {\dfrac{{ - 1}}{2},4} \right)$

The directrix of the parabola is $\left( {3 - 1} \right) = 2$

**Thus the vertex, focus and directrix of the parabola are $\left( {\dfrac{{ - 1}}{2},3} \right)$, $\left( {\dfrac{{ - 1}}{2},4} \right)$ and $2$ respectively.**

**Note:**The parabola has two real foci situated on its axis one of which is the focus S and the other lies at infinity. The corresponding directrix is also at infinity.

A parabola is a curve where any point is at an equal distance from

A fixed point and a fixed straight line

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