Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (- 3, 4); Directrix: y = 2

Verified step by step guidance

1. Recall that the standard form of a parabola with a vertical axis of symmetry is given by the equation \( (x - h)^2 = 4p(y - k) \), where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus (or directrix).
2. Identify the focus \( F(-3, 4) \) and the directrix \( y = 2 \). The vertex \( V \) lies exactly halfway between the focus and the directrix along the vertical line.
3. Calculate the vertex coordinates by finding the midpoint between the focus and the directrix: \( k = \frac{4 + 2}{2} \) for the \( y \)-coordinate, and \( h = -3 \) (same as the focus's \( x \)-coordinate).
4. Determine the value of \( p \), which is the distance from the vertex to the focus (or directrix). Since the parabola opens vertically, \( p = 4 - k \) (distance from vertex to focus).
5. Substitute \( h \), \( k \), and \( p \) into the standard form equation \( (x - h)^2 = 4p(y - k) \) to write the equation of the parabola.

Key Concepts

Definition of a Parabola

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. Understanding this definition helps in deriving the equation of the parabola based on given focus and directrix.

Standard Form of a Parabola

The standard form of a parabola's equation depends on its orientation. For a vertical parabola, it is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix. This form is essential for writing the equation once the vertex and p are known.

Finding the Vertex and Parameter p

The vertex lies midway between the focus and directrix. The distance p is the distance from the vertex to the focus (positive if the parabola opens upward, negative if downward). Calculating these allows you to write the parabola's equation in standard form.