Find the vertex, focus, and directrix of the parabola with the given equation. Then graph the parabola. (x-4)^2 = 4(y+1)

Verified step by step guidance

1. Identify the standard form of the parabola equation. The given equation \((x-4)^2 = 4(y+1)\) is in the form \((x-h)^2 = 4p(y-k)\), which represents a vertical parabola.
2. Determine the vertex \((h, k)\) of the parabola. From the equation \((x-4)^2 = 4(y+1)\), we can see that \(h = 4\) and \(k = -1\). Therefore, the vertex is \((4, -1)\).
3. Find the value of \(p\) from the equation \((x-4)^2 = 4(y+1)\). Here, \(4p = 4\), so \(p = 1\).
4. Calculate the focus of the parabola. Since the parabola opens upwards, the focus is \((h, k + p)\). Substituting the values, the focus is \((4, -1 + 1) = (4, 0)\).
5. Determine the equation of the directrix. The directrix is a horizontal line given by \(y = k - p\). Substituting the values, the directrix is \(y = -1 - 1 = -2\).

Key Concepts

Parabola Definition

A parabola is a symmetric curve formed by the intersection of a cone with a plane parallel to its side. It can be represented by a quadratic equation in the form (x-h)² = 4p(y-k) or (y-k)² = 4p(x-h), where (h, k) is the vertex. Understanding the standard form of a parabola is essential for identifying its key features.

Vertex of a Parabola

The vertex of a parabola is the point where it changes direction, representing either the maximum or minimum point of the curve. For the equation (x-4)² = 4(y+1), the vertex can be found at the point (h, k), which corresponds to (4, -1) in this case. This point is crucial for graphing the parabola accurately.

Focus and Directrix

The focus and directrix are key components that define a parabola's shape. The focus is a fixed point located at (h, k+p), while the directrix is a line given by y = k - p for vertical parabolas. In the given equation, identifying these elements helps in sketching the parabola and understanding its geometric properties.