In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola.8x^2 + 4y = 0

Verified step by step guidance

1. Rewrite the given equation in the standard form of a parabola. Start by isolating the quadratic term: \(8x^2 = -4y\).
2. Divide the entire equation by 8 to simplify: \(x^2 = -\frac{1}{2}y\).
3. Recognize that this equation is in the form \(x^2 = 4py\), where \(4p = -\frac{1}{2}\). Solve for \(p\) to find \(p = -\frac{1}{8}\).
4. Identify the focus of the parabola. Since the parabola opens downwards (negative \(p\)), the focus is at \((0, p) = (0, -\frac{1}{8})\).
5. Determine the directrix of the parabola. The directrix is a horizontal line given by \(y = -p\), so \(y = \frac{1}{8}\).

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 x = ay^2 + by + c. The key features of a parabola include its vertex, focus, and directrix, which help define its shape and position in the coordinate plane.

Focus and Directrix

The focus of a parabola is a fixed point located inside the curve, while the directrix is a line perpendicular to the axis of symmetry of the parabola. The defining property of a parabola is that any point on the curve is equidistant from the focus and the directrix. This relationship is crucial for graphing the parabola and understanding its geometric properties.

Converting to Standard Form

To analyze the given equation of a parabola, it is often necessary to convert it into standard form. The standard form for a vertical parabola is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix. For the equation 8x^2 + 4y = 0, rearranging it will help identify the vertex, focus, and directrix more easily.