To write the equation of a parabola in vertex form, we use the formula:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. Given that the vertex is \((2, -4)\), we can substitute \(h = 2\) and \(k = -4\):
\[ y = a(x - 2)^2 - 4 \]
Next, we need to determine the value of \(a\). We can do this by using the other point through which the parabola passes, which is \((8, \frac{1}{2})\). We substitute \(x = 8\) and \(y = \frac{1}{2}\) into the equation:
\[ \frac{1}{2} = a(8 - 2)^2 - 4 \]
First, simplify \( (8 - 2)^2 \):
\[ (8 - 2)^2 = 6^2 = 36 \]
Now substitute this back into the equation:
\[ \frac{1}{2} = a(36) - 4 \]
Next, solve for \(a\):
\[ \frac{1}{2} + 4 = 36a \]
Convert \(4\) to a fraction:
\[ \frac{1}{2} + \frac{8}{2} = 36a \]
This gives:
\[ \frac{9}{2} = 36a \]
Now, solve for \(a\) by dividing both sides by 36:
\[ a = \frac{9}{2 \times 36} = \frac{9}{72} = \frac{1}{8} \]
Now that we have \(a\), we can write the final equation:
\[ y = \frac{1}{8}(x - 2)^2 - 4 \]
This is the equation of the parabola in vertex form:
\[ \boxed{y = \frac{1}{8}(x - 2)^2 - 4} \]