In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions. Vertex: (2, - 3); Focus: (2, - 5)

Key Concepts

Parabola Definition

A parabola is a symmetric curve formed by the intersection of a cone with a plane parallel to its side. It can open upwards, downwards, left, or right, depending on its orientation. The key features of a parabola include its vertex, focus, and directrix, which help define its shape and position in the coordinate plane.

Vertex and Focus

The vertex of a parabola is the highest or lowest point on the curve, depending on its orientation. The focus is a fixed point located inside the parabola, and it plays a crucial role in defining the parabola's shape. The distance between the vertex and the focus determines the 'width' of the parabola, influencing how quickly it opens.

Standard Form of a Parabola

The standard form of a parabola's equation can be expressed as (x - h)² = 4p(y - k) for vertical parabolas or (y - k)² = 4p(x - h) for horizontal parabolas, where (h, k) is the vertex and p is the distance from the vertex to the focus. This form allows for easy identification of the vertex and focus, facilitating graphing and analysis of the parabola's properties.