In Exercises 17–30, 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. insert step 1: Identify the vertex and focus of the parabola. The vertex is (2, -3) and the focus is (2, -5).
2. insert step 2: Determine the orientation of the parabola. Since the x-coordinates of the vertex and focus are the same, the parabola opens vertically.
3. insert step 3: Calculate the distance between the vertex and the focus, which is the value of 'p'. Here, p = -2 because the focus is below the vertex.
4. insert step 4: Use the vertex form of a vertical parabola equation: (x - h)^2 = 4p(y - k), where (h, k) is the vertex.
5. insert step 5: Substitute the vertex (h, k) = (2, -3) and p = -2 into the equation to get the standard form.

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.