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. Identify the form of the given equation. The equation \((y + 3)^2 = 12(x + 1)\) is in the form \((y - k)^2 = 4p(x - h)\), which represents a parabola that opens either to the right or left, with vertex at \((h, k)\).
2. From the equation, determine the vertex by comparing it to the standard form. Here, \(h = -1\) and \(k = -3\), so the vertex is at \((-1, -3)\).
3. Find the value of \$4p\( by comparing the equation. Since \)4p = 12\(, solve for \)p\( to get \)p = 3\(. The positive value of \)p$ indicates the parabola opens to the right.
4. Locate the focus using the vertex and \(p\). For a parabola opening right, the focus is at \((h + p, k)\), so substitute the values to find the focus.
5. Write the equation of the directrix, which is a vertical line for this parabola. The directrix is located at \(x = h - p\), so substitute the values to find its equation.

Key Concepts

Standard Form of a Parabola

A parabola can be expressed in standard form as (y - k)^2 = 4p(x - h) for horizontal parabolas, where (h, k) is the vertex. This form helps identify the vertex and the direction the parabola opens. Recognizing and rewriting the equation in this form is essential for further analysis.

Vertex, Focus, and Directrix of a Parabola

The vertex is the parabola's turning point, located at (h, k). The focus lies p units from the vertex inside the parabola, and the directrix is a line p units opposite the focus. The value of p determines the distance from the vertex to the focus and directrix, crucial for graphing.

Graphing Parabolas

Graphing involves plotting the vertex, focus, and directrix, then sketching the curve opening toward the focus. Understanding the orientation (horizontal or vertical) and the distance p helps accurately draw the parabola's shape and position on the coordinate plane.