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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1. Identify the standard form of the equation for a parabola that opens left or right: \((y - k)^2 = 4p(x - h)\), where \(h\), \(k\) are the coordinates of the vertex, and \(p\) is the distance from the vertex to the focus, and also to the directrix.
2. Compare the given equation \( (y - 1)^2 = 4(x - 1) \) with the standard form to find the vertex \( (h, k) \) of the parabola.
3. Determine the value of \(p\) from the equation by comparing it to the standard form. This will help in finding the focus and the directrix of the parabola.
4. Calculate the coordinates of the focus using \( (h + p, k) \) since the parabola opens to the right (positive \(p\)).
5. Find the equation of the directrix, which is a vertical line \(x = h - p\) since the parabola opens to the right.

Key Concepts

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 - k)^2 = 4p(x - h) for horizontal parabolas, where (h, k) is the vertex and p is the distance from the vertex to the focus.

Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction, representing either the maximum or minimum point of the parabola. For the equation (y - 1)^2 = 4(x - 1), the vertex can be identified as (1, 1), which is derived from the standard form of the parabola.

Focus and Directrix

The focus of a parabola is a fixed point located inside the curve, while the directrix is a line outside the curve. For the equation given, the focus can be found at (1 + p, 1) and the directrix is the line x = 1 - p, where p is the distance from the vertex to the focus, determined by the coefficient in the equation.