In Exercises 35–42, find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola.(y + 3)^2 = 12(x + 1)

Verified step by step guidance

1. Rewrite the given equation \((y + 3)^2 = 12(x + 1)\) in the standard form of a parabola \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex.
2. Identify the vertex \((h, k)\) from the equation. Here, \(h = -1\) and \(k = -3\), so the vertex is \((-1, -3)\).
3. Determine the value of \(p\) by comparing \(12\) with \(4p\). Solve \(4p = 12\) to find \(p = 3\).
4. Find the focus of the parabola using the vertex \((-1, -3)\) and \(p = 3\). Since the parabola opens to the right, the focus is \((h + p, k) = (-1 + 3, -3) = (2, -3)\).
5. Determine the directrix of the parabola, which is a vertical line \(x = h - p\). Substitute \(h = -1\) and \(p = 3\) to get the directrix \(x = -1 - 3 = -4\).

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 be represented by a quadratic equation in the form y = ax^2 + bx + c or in vertex form. The key features of a parabola include its vertex, focus, and directrix, which help define its shape and position in the coordinate plane.

Vertex Form of a Parabola

The vertex form of a parabola is expressed as (y - k)^2 = 4p(x - h), where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix. This form allows for easy identification of the vertex and the direction in which the parabola opens. In the given equation, identifying the vertex is crucial for graphing the parabola accurately.

Focus and Directrix

The focus of a parabola is a fixed point located along the axis of symmetry, while the directrix is a line perpendicular to this axis. The distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. Understanding these concepts is essential for determining the geometric properties of the parabola and for accurately graphing it.