Find the standard form of the equation of each parabola satisfying the given conditions. Vertex: (2, - 3); Focus: (2, - 5)

Verified step by step guidance

1. Identify the vertex \( V = (2, -3) \) and the focus \( F = (2, -5) \). Since the x-coordinates are the same, the parabola opens vertically (either up or down).
2. Calculate the distance \( p \) between the vertex and the focus. This distance determines how far the parabola opens from the vertex. Use the formula \( p = y_{focus} - y_{vertex} \).
3. Write the standard form of the parabola that opens vertically: \( (x - h)^2 = 4p(y - k) \), where \( (h, k) \) is the vertex.
4. Substitute the vertex coordinates \( (h, k) = (2, -3) \) and the value of \( p \) found in step 2 into the equation.
5. Simplify the equation to get the standard form 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 when the vertex and focus are known.

Standard Form of a Parabola

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

Finding the Parameter p

The parameter p represents the distance from the vertex to the focus (or directrix) and determines the parabola's width and direction. Calculating p from the given vertex and focus coordinates is essential to write the equation in standard form.