To determine the values of [tex]\( a \)[/tex], [tex]\( h \)[/tex], and [tex]\( k \)[/tex] in the vertex form of the parabola, we will analyze the given information about the vertex and the directrix.
1. Identify [tex]\( h \)[/tex] and [tex]\( k \)[/tex] from the vertex:
The vertex [tex]\(\left(-\frac{1}{2}, 3\right)\)[/tex] gives us:
[tex]\[
h = -\frac{1}{2}
\][/tex]
[tex]\[
k = 3
\][/tex]
2. Use the directrix to find [tex]\( a \)[/tex]:
The directrix given is [tex]\( x = -\frac{13}{24} \)[/tex]. To find the value of [tex]\( a \)[/tex] in the equation of the parabola, consider the distance from the directrix to the vertex. For a vertically oriented parabola, we use the relationship:
[tex]\[
| \text{directrix} - h | = \frac{1}{4a}
\][/tex]
Here, [tex]\(| \text{directrix} - h |\)[/tex] is the absolute value of the difference between the x-coordinates of the directrix and the vertex.
The x-coordinate of the vertex is:
[tex]\[
h = -\frac{1}{2}
\][/tex]
The x-coordinate of the directrix is:
[tex]\[
\text{directrix} = -\frac{13}{24}
\][/tex]
Calculate the absolute difference:
[tex]\[
\left| -\frac{13}{24} - \left( -\frac{1}{2} \right) \right| = \left| -\frac{13}{24} + \frac{12}{24} \right| = \left| -\frac{13}{24} + \frac{12}{24} \right| = \left| -\frac{1}{24} \right| = \frac{1}{24}
\][/tex]
Now, use this difference to find [tex]\( a \)[/tex]:
[tex]\[
\frac{1}{24} = \frac{1}{4a}
\][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[
4a = 24 \implies a = 6
\][/tex]
Therefore, we find:
[tex]\[
h = -\frac{1}{2}, \quad k = 3, \quad a = 6
\][/tex]
So, the values for [tex]\( a \)[/tex], [tex]\( h \)[/tex], and [tex]\( k \)[/tex] in the vertex form of the parabola are:
[tex]\[
a = 6, \quad h = -\frac{1}{2}, \quad k = 3
\][/tex]
