Find the vertex, focus, and directrix of the parabola with the given equation. Then graph the parabola. x^2 - 4x - 2y = 0

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Rewrite the given equation in standard form for a parabola. Start by isolating the terms involving x on one side of the equation: \(x^2 - 4x = 2y\).

Complete the square for the \(x\)-terms to rewrite the equation in vertex form. Take half the coefficient of \(x\) (which is \(-4\)), square it, and add it to both sides of the equation. This gives \(x^2 - 4x + 4 = 2y + 4\).

Simplify the equation to express it in vertex form: \((x - 2)^2 = 2(y + 2)\). This shows the parabola is vertical, opening upwards, with its vertex at \((2, -2)\).

Identify the focus and directrix. For a vertical parabola in the form \((x - h)^2 = 4p(y - k)\), the focus is \((h, k + p)\) and the directrix is \(y = k - p\). Here, \(4p = 2\), so \(p = \frac{1}{2}\). The focus is \((2, -2 + \frac{1}{2}) = (2, -\frac{3}{2})\), and the directrix is \(y = -2 - \frac{1}{2} = -\frac{5}{2}\).

Graph the parabola by plotting the vertex \((2, -2)\), the focus \((2, -\frac{3}{2})\), and the directrix \(y = -\frac{5}{2}\). Sketch the parabola opening upwards, ensuring it is symmetric about the vertical line passing through the vertex.

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주요 개념

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Parabola Definition

A parabola is a symmetric curve formed by the intersection of a cone with a plane parallel to its side. In algebra, it can be represented by a quadratic equation in the form y = ax^2 + bx + c or x = ay^2 + by + c. The key features of a parabola include its vertex, focus, and directrix, which help define its shape and position in the coordinate plane.

Vertex of a Parabola

The vertex of a parabola is the point where it changes direction, representing either the maximum or minimum value of the quadratic function. For a parabola given in standard form, the vertex can be found using the formula (-b/2a, f(-b/2a)) for y = ax^2 + bx + c. In the context of the given equation, identifying the vertex is crucial for graphing the parabola accurately.

Focus and Directrix

The focus and directrix are essential components of a parabola that define its geometric properties. The focus is a fixed point inside the parabola where all reflected lines converge, while the directrix is a line perpendicular to the axis of symmetry of the parabola. The distance from any point on the parabola to the focus is equal to the distance from that point to the directrix, which is fundamental for understanding the parabola's shape and orientation.