In Exercises 31–34, find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d).(x + 1)^2 = - 4(y + 1)

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Identify the standard form of the equation for a parabola that opens upwards or downwards: \((x - h)^2 = 4p(y - k)\), where \((h, k)\) is the vertex of the parabola, and \(p\) is the distance from the vertex to the focus and from the vertex to the directrix.

Rewrite the given equation \((x + 1)^2 = -4(y + 1)\) in the standard form by comparing it with the standard form. Here, \(h = -1\), \(k = -1\), and \(4p = -4\).

Solve for \(p\) by dividing the coefficient of \((y + 1)\) by 4: \(p = -1\). This tells us that the parabola opens downwards because \(p\) is negative.

Calculate the coordinates of the vertex using the values of \(h\) and \(k\) obtained from the equation. The vertex is at \((-1, -1)\).

Determine the focus and directrix of the parabola. The focus is located at \((-1, -1 - p)\) or \((-1, -2)\), and the directrix is the horizontal line \(y = -1 + p\) or \(y = 0\).

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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. In algebra, parabolas can be represented by quadratic equations, typically in the form y = ax^2 + bx + c or in vertex form. The orientation of the parabola (opening upwards, downwards, left, or right) is determined by the coefficients in the equation.

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 the equation in the form (x - h)^2 = 4p(y - k), the vertex is located at the point (h, k). In the given equation, identifying the vertex is crucial for graphing the parabola accurately.

Focus and Directrix

The focus and directrix are key components that define a parabola. The focus is a fixed point inside the parabola where all lines drawn parallel to the axis of symmetry reflect through, while the directrix is a line perpendicular to the axis of symmetry. The distance from any point on the parabola to the focus is equal to the distance from that point to the directrix, which is essential for understanding the geometric properties of the parabola.

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