To find the values for [tex]\(a\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex] in the vertex form of the parabola [tex]\(x = a(y - k)^2 + h\)[/tex], we will proceed step by step.
### Step 1: Identify the Vertex ([tex]\(h\)[/tex], [tex]\(k\)[/tex])
The given vertex is [tex]\(\left(-\frac{1}{2}, 3\right)\)[/tex]. This gives us the values for [tex]\(h\)[/tex] and [tex]\(k\)[/tex]:
[tex]\[ h = -\frac{1}{2} \][/tex]
[tex]\[ k = 3 \][/tex]
### Step 2: Determine the Directrix
The directrix is given as [tex]\(x = -\frac{13}{24}\)[/tex].
### Step 3: Calculate the Distance from the Vertex to the Directrix
The distance from the vertex to the directrix is the absolute value of the difference between the [tex]\(x\)[/tex]-coordinate of the vertex and the [tex]\(x\)[/tex]-coordinate of the directrix:
[tex]\[ \text{distance to directrix} = \left| -\frac{1}{2} - \left( -\frac{13}{24} \right) \right| \][/tex]
### Step 4: Simplify the Distance Calculation
First, convert [tex]\(-\frac{1}{2}\)[/tex] to a fraction with denominator 24:
[tex]\[ -\frac{1}{2} = -\frac{12}{24} \][/tex]
Now, perform the subtraction:
[tex]\[ \text{distance to directrix} = \left| -\frac{12}{24} + \frac{13}{24} \right| = \left| \frac{1}{24} \right| = \frac{1}{24} \][/tex]
### Step 5: Calculate [tex]\(a\)[/tex]
For a parabola that opens horizontally, the formula for the distance from the vertex to the directrix is [tex]\(\frac{1}{4|a|}\)[/tex]. Since we have the distance to the directrix, we can solve for [tex]\(|a|\)[/tex]:
[tex]\[ \frac{1}{4|a|} = \frac{1}{24} \][/tex]
[tex]\[ 4|a| = 24 \][/tex]
[tex]\[ |a| = \frac{24}{4} = 6 \][/tex]
Since the parabola opens to the right, the value of [tex]\(a\)[/tex] should be positive:
[tex]\[ a = 6 \][/tex]
### Final Answer
The values for [tex]\(a\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex] are:
[tex]\[ a = 6 \][/tex]
[tex]\[ h = -\frac{1}{2} \][/tex]
[tex]\[ k = 3 \][/tex]
