Find the focus and directrix of the parabola with the given equation. Then graph the parabola. x² = - 16y

Verified step by step guidance

1. Rewrite the given equation \(x^2 = -16y\) in the standard form of a vertical parabola, which is \(x^2 = 4py\). Identify the value of \$4p$ by comparing the two equations.
2. From the equation \(x^2 = 4py\), solve for \(p\) by dividing both sides by 4, so \(p = \frac{-16}{4}\). This value of \(p\) represents the distance from the vertex to the focus and from the vertex to the directrix.
3. Determine the vertex of the parabola. Since the equation is in the form \(x^2 = 4py\), the vertex is at the origin \((0,0)\).
4. Find the focus of the parabola. For a vertical parabola \(x^2 = 4py\), the focus is located at \((0, p)\). Use the value of \(p\) found in step 2 to write the coordinates of the focus.
5. Find the equation of the directrix. The directrix is a horizontal line given by \(y = -p\). Use the value of \(p\) to write the equation of the directrix.

Key Concepts

Standard Form of a Parabola

The standard form of a parabola that opens vertically is x² = 4py, where p represents the distance from the vertex to the focus. Recognizing this form helps identify the parabola's orientation and key features such as the vertex, focus, and directrix.

Focus and Directrix of a Parabola

The focus is a fixed point inside the parabola, and the directrix is a line perpendicular to the axis of symmetry. For x² = 4py, the focus is at (0, p) and the directrix is y = -p. These elements define the parabola's shape and are essential for graphing.

Graphing Parabolas

Graphing involves plotting the vertex, focus, and directrix, then sketching the curve that is equidistant from the focus and directrix. Understanding the parabola's orientation and key points allows accurate representation of its shape on the coordinate plane.