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

검증된 단계별 안내

Rewrite the given equation in standard form for a parabola. The equation \((x-h)^2 = 4p(y-k)\) represents a parabola that opens vertically, where \((h, k)\) is the vertex and \(p\) determines the distance from the vertex to the focus and directrix. In this case, \((x-4)^2 = 4(y+1)\) is already in standard form with \(h = 4\), \(k = -1\), and \(4p = 4\).

Identify the vertex of the parabola. From the equation, the vertex is \((h, k) = (4, -1)\).

Determine the value of \(p\). Since \(4p = 4\), divide both sides by 4 to find \(p = 1\). This means the focus is 1 unit away from the vertex along the axis of symmetry, and the directrix is 1 unit away in the opposite direction.

Find the focus. Since the parabola opens vertically, the focus is located \(p\) units above the vertex. Add \(p = 1\) to the \(y\)-coordinate of the vertex: \((4, -1 + 1) = (4, 0)\).

Find the directrix. The directrix is a horizontal line \(p\) units below the vertex. Subtract \(p = 1\) from the \(y\)-coordinate of the vertex: \(y = -1 - 1 = -2\). The directrix is the line \(y = -2\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 and p is the distance from the vertex to the focus or directrix.

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 in this case is (4, -1). This point is crucial for graphing the parabola.

Focus and Directrix

The focus of a parabola is a fixed point located along the axis of symmetry, while the directrix is a line perpendicular to this axis. The distance from the vertex to the focus is equal to the distance from the vertex to the directrix, denoted as 'p'. For the given equation, the focus is at (4, -1 + p) and the directrix is the line y = -1 - p, where p is derived from the equation.