Let's start by identifying important components from the given information:
1. Focus: The focus of the parabola is [tex]\(\left(-\frac{3}{2}, -\frac{11}{4}\right)\)[/tex].
2. Directrix: The directrix is given as [tex]\( y = -\frac{21}{4} \)[/tex].
### Step 1: Determine the Vertex
The vertex of a parabola that is vertical (i.e., opens upwards or downwards) lies midway between the focus and the directrix. The y-coordinate of the vertex, [tex]\( k \)[/tex], is the average of the y-coordinates of the focus and the directrix.
So, let's calculate the y-coordinate of the vertex:
[tex]\[ k = \frac{-\frac{11}{4} + \left(-\frac{21}{4}\right)}{2} = \frac{-\frac{11}{4} - \frac{21}{4}}{2} = \frac{-\frac{32}{4}}{2} = \frac{-8}{2} = -4 \][/tex]
The x-coordinate of the vertex, [tex]\( h \)[/tex], is the same as that of the focus:
[tex]\[ h = -\frac{3}{2} \][/tex]
Thus, the vertex of the parabola is [tex]\(\left(-\frac{3}{2}, -4 \right)\)[/tex].
### Step 2: Determine the Distance [tex]\( p \)[/tex]
The distance [tex]\( p \)[/tex] from the vertex to the focus (which is also the distance from the vertex to the directrix) is the absolute difference between [tex]\( k \)[/tex] and the y-coordinate of the directrix.
[tex]\[ p = \left| k - \left(-\frac{21}{4}\right) \right| = \left| -4 - \left(-\frac{21}{4}\right) \right| = \left| -4 + \frac{21}{4} \right| = \left| \frac{-16 + 21}{4} \right| = \left| \frac{5}{4} \right| = \frac{5}{4} \][/tex]
### Step 3: Write the Equation in Vertex Form
For a vertically oriented parabola, the vertex form of the equation is given by:
[tex]\[ (x - h)^2 = 4p(y - k) \][/tex]
Substituting the vertex coordinates [tex]\((h, k) = \left(-\frac{3}{2}, -4\right)\)[/tex] and [tex]\( p = \frac{5}{4} \)[/tex]:
[tex]\[ (x - \left(-\frac{3}{2}\right))^2 = 4 \cdot \frac{5}{4} (y - (-4)) \][/tex]
Simplifying this gives us:
[tex]\[ (x + \frac{3}{2})^2 = 5(y + 4) \][/tex]
Thus, the vertex form of the equation for the given parabola is:
[tex]\[ (x + \frac{3}{2})^2 = 5(y + 4) \][/tex]
