Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. (y + 1)² = - 8x

Verified step by step guidance

1. Identify the form of the given equation. The equation \((y + 1)^2 = -8x\) is a parabola in the form \((y - k)^2 = 4p(x - h)\), which represents a parabola that opens either left or right.
2. Rewrite the equation to match the standard form: \((y - (-1))^2 = -8(x - 0)\). Here, the vertex is at \((h, k) = (0, -1)\).
3. Determine the value of \(p\) by comparing \$4p\( to the coefficient of \)(x - h)\(. Since \)4p = -8\(, solve for \)p\( to get \)p = -2$. The negative sign indicates the parabola opens to the left.
4. Find the focus using the vertex and \(p\). The focus lies \(p\) units from the vertex along the axis of symmetry. Since the parabola opens left, the focus is at \((h + p, k) = (0 - 2, -1)\).
5. Find the directrix, which is a vertical line \(p\) units in the opposite direction from the vertex. The directrix is the line \(x = h - p = 0 - (-2) = 2\).

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, direction, and width of the parabola. Recognizing and rewriting the equation into this form is essential for further analysis.

Vertex, Focus, and Directrix of a Parabola

The vertex is the parabola's turning point, the focus is a fixed point inside the curve, and the directrix is a line outside the curve. The distance from the vertex to the focus and directrix is |p|, where p determines the parabola's shape and orientation. These elements define the parabola's geometry.

Graphing Parabolas

Graphing involves plotting the vertex, focus, and directrix, then sketching the curve symmetric about the axis through the vertex and focus. Understanding the parabola's orientation (opening left, right, up, or down) and scale based on |4p| helps create an accurate graph.