Given the vertex [tex]\(\left(-\frac{1}{2}, 3\right)\)[/tex] and the directrix [tex]\(x=-\frac{13}{24}\)[/tex], what are the values for [tex]\(a\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex] in the vertex form of the parabola?



Answer :

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]