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

Verified step by step guidance

1. Identify the form of the given equation. The equation \((x - 2)^2 = 8(y - 1)\) is in the form \((x - h)^2 = 4p(y - k)\), which represents a parabola that opens either upward or downward, where \((h, k)\) is the vertex.
2. From the equation, determine the vertex \((h, k)\). Here, \(h = 2\) and \(k = 1\), so the vertex is at the point \((2, 1)\).
3. Find the value of \(p\) by comparing \$4p\( to the coefficient on the right side. Since \)4p = 8\(, solve for \)p\( to get \)p = 2\(. The positive value of \)p$ indicates the parabola opens upward.
4. Locate the focus using the vertex and \(p\). For a parabola opening upward, the focus is at \((h, k + p)\), so substitute to get the focus at \((2, 1 + 2)\).
5. Write the equation of the directrix, which is a horizontal line \(p\) units opposite the focus from the vertex. For this parabola, the directrix is \(y = k - p\), so substitute to get \(y = 1 - 2\).

Key Concepts

Standard Form of a Parabola

The standard form of a parabola that opens vertically is (x - h)^2 = 4p(y - k), where (h, k) is the vertex. This form helps identify the vertex directly and the value of p, which determines the distance from the vertex to the focus and directrix.

Vertex, Focus, and Directrix

The vertex is the parabola's turning point. The focus is a point inside the parabola where all reflected rays converge, located p units from the vertex along the axis of symmetry. The directrix is a line p units from the vertex on the opposite side of the focus, serving as a reference for defining the parabola.

Graphing a Parabola

Graphing involves plotting the vertex, focus, and directrix, then sketching the curve symmetric about the axis through the vertex and focus. Knowing the direction the parabola opens (up or down) depends on the sign of p in the equation, guiding the shape and orientation of the graph.