In Exercises 35–42, find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. (x + 1)^2 = - 8(y + 1)

Verified step by step guidance

1. Rewrite the given equation \((x + 1)^2 = -8(y + 1)\) in the standard form of a parabola. The standard form for a vertical parabola is \((x - h)^2 = 4p(y - k)\).
2. Identify the vertex \((h, k)\) from the equation. Here, \(h = -1\) and \(k = -1\), so the vertex is \((-1, -1)\).
3. Determine the value of \(4p\) from the equation. In this case, \(4p = -8\), which means \(p = -2\).
4. Find the focus of the parabola. Since \(p = -2\) and the parabola opens downwards, the focus is \((-1, -1 - 2)\), which simplifies to \((-1, -3)\).
5. Determine the directrix of the parabola. The directrix is a horizontal line \(y = k - p\). Substitute \(k = -1\) and \(p = -2\) to find the directrix: \(y = -1 + 2 = 1\).

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 of a Parabola

The vertex of a parabola is the point where it changes direction, representing either the maximum or minimum point of the curve. For the equation (x + 1)^2 = -8(y + 1), the vertex can be found by rewriting the equation in vertex form, which reveals the coordinates of the vertex directly. In this case, the vertex is at the point (-1, -1).

Focus and Directrix

The focus of a parabola is a fixed point located inside the curve, while the directrix is a line outside the curve. The distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. For the given equation, the focus and directrix can be determined using the standard form of a parabola, which relates these elements to the coefficients in the equation.