Find the focus and directrix of the parabola with the given equation. Then graph the parabola. x² = 12y

Verified step by step guidance

1. Identify the form of the given parabola equation. The equation is \(x^2 = 12y\), which matches the standard form of a vertical parabola: \(x^2 = 4py\), where the vertex is at the origin \((0,0)\) and the parabola opens upward if \(p > 0\) or downward if \(p < 0\).
2. Compare the given equation \(x^2 = 12y\) with the standard form \(x^2 = 4py\) to find the value of \(p\). Set \$4p = 12\(, then solve for \)p$.
3. Use the value of \(p\) to determine the focus of the parabola. For a parabola in the form \(x^2 = 4py\), the focus is located at \((0, p)\).
4. Find the equation of the directrix. The directrix is a horizontal line given by \(y = -p\) for this form of parabola.
5. Summarize the results: the vertex is at the origin \((0,0)\), the focus is at \((0, p)\), and the directrix is the line \(y = -p\). Use these to sketch the parabola, noting it opens upward since \(p\) is positive.

Key Concepts

Standard Form of a Parabola

A parabola can be expressed in standard form as x² = 4py or y² = 4px, where p represents the distance from the vertex to the focus and directrix. Recognizing the equation's form helps identify the parabola's orientation and key features.

Focus and Directrix of a Parabola

The focus is a fixed point inside the parabola, and the directrix is a line outside it, both equidistant from the vertex. For x² = 4py, the focus is at (0, p) and the directrix is the line y = -p, which guide the parabola's shape.

Graphing a Parabola

Graphing involves plotting the vertex, focus, and directrix, then sketching the curve that is equidistant from the focus and directrix. Understanding symmetry and the parabola's opening direction (upward or downward) is essential for accurate graphing.