To analyze parabolas, we utilize the standard equation of a parabola, which can be expressed as \(4p(y - k) = (x - h)^2\). In this equation, \((h, k)\) represents the vertex of the parabola, while \(p\) indicates the distance from the vertex to the focus and the directrix. If the equation lacks \(h\) and \(k\), the parabola is centered at the origin, simplifying the equation to \(4py = x^2\).

For example, consider a parabola represented by \(4py = x^2\). Here, since there are no \(h\) and \(k\) values, the vertex is at the origin \((0, 0)\). Setting \(4p = 16\) allows us to solve for \(p\), yielding \(p = 4\). A positive \(p\) indicates that the parabola opens upward. Consequently, the focus is located at \((0, 4)\) and the directrix is the line \(y = -4\).

In another case, if we have a parabola defined by \(4py = \frac{1}{3}\), we again identify the vertex at the origin. Setting \(4p = \frac{1}{3}\) leads to \(p = \frac{1}{12}\). The positive value of \(p\) confirms that the parabola opens upward, placing the focus at \((0, \frac{1}{12})\) and the directrix at \(y = -\frac{1}{12}\).

When the equation includes \(h\) and \(k\), such as \(4p(y - 1) = (x - 2)^2\), the vertex shifts to \((2, 1)\). Here, setting \(4p = 8\) gives \(p = 2\). The focus, therefore, is at \((2, 3)\) (moving up 2 units from the vertex), while the directrix is the line \(y = -1\) (2 units below the vertex).

For a parabola like \(-12y = (x + 1)^2\), we first rewrite it to identify the vertex. The vertex is at \((-1, 0)\) since \(h = -1\) and \(k = 0\). Setting \(4p = -12\) results in \(p = -3\), indicating that the parabola opens downward. Thus, the focus is at \((-1, -3)\) and the directrix is the line \(y = 3\) (3 units above the vertex).

In summary, the vertex, focus, and directrix of a parabola can be determined using the standard form of the equation. The value of \(p\) not only indicates the direction in which the parabola opens but also helps in locating the focus and directrix relative to the vertex. Understanding these relationships is crucial for solving problems involving parabolas effectively.