Find the vertex, focus, and directrix of the parabola with the given equation. Then graph the parabola. (y-2)^2 = -16x

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Rewrite the given equation \((y - 2)^2 = -16x\) in its standard form for a parabola. The standard form for a parabola that opens left or right is \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex and \(p\) determines the distance and direction of the focus and directrix.

Compare \((y - 2)^2 = -16x\) with \((y - k)^2 = 4p(x - h)\). From this comparison, identify \(h = 0\), \(k = 2\), and \(4p = -16\). Solve for \(p\) by dividing \(-16\) by \(4\), which gives \(p = -4\).

The vertex of the parabola is \((h, k) = (0, 2)\). Since \(p = -4\), the parabola opens to the left (negative \(x\)-direction). The focus is located \(p\) units to the left of the vertex, so the focus is at \((h + p, k) = (0 - 4, 2) = (-4, 2)\).

The directrix is a vertical line \(p\) units to the right of the vertex. Since \(p = -4\), the directrix is at \(x = h - p = 0 - (-4) = 4\).

To graph the parabola, plot the vertex \((0, 2)\), the focus \((-4, 2)\), and the directrix \(x = 4\). Sketch the parabola opening to the left, ensuring it is symmetric about the horizontal line \(y = 2\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 y = ax^2 + bx + c or in vertex form. The standard form of a parabola that opens horizontally is (y-k)^2 = 4p(x-h), where (h, k) is the vertex and p is the distance from the vertex to the focus.

Vertex, Focus, and Directrix

The vertex of a parabola is the point where it changes direction, while the focus is a fixed point inside the parabola that determines its shape. The directrix is a line perpendicular to the axis of symmetry of the parabola, and the distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. For the equation (y-2)^2 = -16x, the vertex is at (0, 2), the focus is at (-4, 2), and the directrix is the line x = 4.

Graphing Parabolas

Graphing a parabola involves plotting its vertex, focus, and directrix, and then sketching the curve that opens towards the focus. The orientation of the parabola (upward, downward, left, or right) is determined by the sign of the coefficient in the standard form equation. For the given equation, since it opens to the left, the graph will reflect this orientation, showing the vertex at (0, 2) and the focus at (-4, 2).

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