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).(y - 1)^2 = - 4(x - 1)

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Step 1: Identify the form of the equation. The given equation is in the form of $(y - k)^2 = 4p(x - h)$, where $(h, k)$ is the vertex of the parabola. In this case, the equation is $(y - 1)^2 = -4(x - 1)$, so the vertex is at $(h, k) = (1, 1)$.

Step 2: Determine the value of $p$. In the standard form of the equation, $4p$ is the coefficient of $(x - h)$ or $(y - k)$. Here, $4p = -4$, so $p = -1$.

Step 3: Find the focus of the parabola. For a parabola in the form $(y - k)^2 = 4p(x - h)$, the focus is at $(h + p, k)$. So, the focus is at $(1 - 1, 1) = (0, 1)$.

Step 4: Determine the equation of the directrix. For a parabola in the form $(y - k)^2 = 4p(x - h)$, the directrix is the vertical line $x = h - p$. So, the directrix is the line $x = 1 - (-1) = 2$.

Step 5: Match the equation to one of the given graphs. The graph of the parabola opens to the left (since $p$ is negative), has its vertex at $(1, 1)$, its focus at $(0, 1)$, and its directrix at $x = 2$. Look for a graph that matches these characteristics.

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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 standard form of a parabola can also be expressed as (y - k)^2 = 4p(x - h), where (h, k) is the vertex and p determines the distance from the vertex to the focus and directrix.

Vertex, Focus, and Directrix

The vertex of a parabola is the highest or lowest point on the curve, depending on its orientation. The focus is a fixed point located inside the parabola, and the directrix is a line perpendicular to the axis of symmetry, equidistant from the vertex as the focus. For the equation given, the vertex can be found directly from the equation, while the focus and directrix can be derived using the value of p.

Graphing Parabolas

Graphing a parabola involves plotting its vertex, focus, and directrix to understand its shape and orientation. The orientation of the parabola (opening left, right, up, or down) is determined by the sign of the coefficient in the equation. For the given equation, the negative sign indicates that the parabola opens to the left, which is crucial for matching it to the correct graph among the options provided.

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