To find the values for [tex]\(a\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex] in the vertex form of the parabola given the vertex [tex]\(\left(-\frac{1}{2}, 3\right)\)[/tex] and the directrix [tex]\(x = -\frac{13}{24}\)[/tex], follow these steps:
1. Identify the vertex coordinates (h, k):
From the given vertex [tex]\(\left(-\frac{1}{2}, 3\right)\)[/tex]:
[tex]\[
h = -\frac{1}{2}, \quad k = 3
\][/tex]
2. Determine the value of [tex]\(a\)[/tex]:
The directrix is given as [tex]\(x = -\frac{13}{24}\)[/tex]. For a parabola with vertex form [tex]\(x = a(y - k)^2 + h\)[/tex], the value of [tex]\(a\)[/tex] can be found using the relationship between the vertex and the directrix.
Since the distance from the vertex to the directrix is [tex]\(|h - \text{directrix}|\)[/tex], this distance is:
[tex]\[
\left|-\frac{1}{2} + \frac{13}{24}\right|
\][/tex]
This simplifies to:
[tex]\[
\left|-\frac{12}{24} + \frac{13}{24}\right| = \left|\frac{1}{24}\right| = \frac{1}{24}
\][/tex]
The value of [tex]\(a\)[/tex] is:
[tex]\[
a = \frac{1}{4 \cdot \left(\frac{1}{24}\right)} = \frac{1}{\frac{1}{6}} = 6
\][/tex]
Therefore, the values are:
[tex]\[
a = 6, \quad h = -0.5, \quad k = 3
\][/tex]
So, the correct answer is [tex]\(a = 6\)[/tex], [tex]\(h = -0.5\)[/tex], and [tex]\(k = 3\)[/tex].
