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)
a. b. c. d.

Verified step by step guidance

1. Identify the form of the given equation. The equation \((x + 1)^2 = -4(y + 1)\) is in the form \((x - h)^2 = 4p(y - k)\), which represents a vertical parabola that opens either up or down.
2. From the equation, determine the vertex \((h, k)\). Here, \(h = -1\) and \(k = -1\), so the vertex is at \((-1, -1)\).
3. Find the value of \(p\) by comparing the equation to the standard form. Since \$4p = -4\(, solve for \)p\( to get \)p = -1$. The negative value indicates the parabola opens downward.
4. Use the vertex and \(p\) to find the focus. The focus lies \(p\) units from the vertex along the axis of symmetry. Since the parabola opens vertically, the focus is at \((h, k + p)\), which is \((-1, -1 - 1)\).
5. Find the directrix, which is a horizontal line \(p\) units in the opposite direction from the vertex. The directrix is the line \(y = k - p\), or \(y = -1 - (-1)\).

Key Concepts

Standard Form of a Parabola

The standard form of a parabola's equation helps identify its orientation and key features. For vertical parabolas, the form is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p indicates the distance from the vertex to the focus and directrix. Recognizing this form allows you to extract important information directly from the equation.

Vertex, Focus, and Directrix of a Parabola

The vertex is the parabola's turning point, given by (h, k). The focus lies inside the parabola at a distance p from the vertex along the axis of symmetry. The directrix is a line perpendicular to the axis of symmetry, located p units opposite the focus. These elements define the parabola's shape and position.

Graph Matching Using Parabola Features

Matching an equation to a graph involves using the vertex, focus, and directrix to determine the parabola's orientation and position. By calculating these features from the equation, you can compare them to the labeled graphs and identify the correct match based on shape, direction (up/down/left/right), and location.