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)
a. b. c. d.

Verified step by step guidance

1. Identify the form of the given equation. The equation \(\left(y - 1\right)^2 = 4\left(x - 1\right)\) is in the form \(\left(y - k\right)^2 = 4p\left(x - h\right)\), which represents a parabola that opens either to the right or left.
2. Determine the vertex of the parabola. The vertex is given by the point \((h, k)\), so here the vertex is at \((1, 1)\).
3. Find the value of \(p\) by comparing the equation to the standard form. Since \$4p = 4\(, it follows that \)p = 1\(. The sign of \)p\( indicates the direction the parabola opens: positive \)p$ means it opens to the right.
4. Calculate the focus. For a parabola in this form, the focus is located at \((h + p, k)\), so substitute \(h = 1\), \(k = 1\), and \(p = 1\) to find the focus.
5. Determine the equation of the directrix. The directrix is a vertical line given by \(x = h - p\). Substitute \(h = 1\) and \(p = 1\) to write the directrix equation.

Key Concepts

Standard Form of a Parabola

The standard form of a parabola's equation helps identify its orientation and key features. For a parabola that opens horizontally, the form is (y - k)^2 = 4p(x - h), where (h, k) is the vertex and p determines the distance to the focus and directrix.

Vertex, Focus, and Directrix

The vertex is the parabola's turning point, given by (h, k). The focus lies p units from the vertex along the axis of symmetry, inside the curve. The directrix is a line p units from the vertex on the opposite side of the focus, serving as a reference line for the parabola.

Graph Matching Using Parabola Features

To match the equation to a graph, use the vertex, focus, and directrix to determine the parabola's shape and position. Identifying these features allows comparison with labeled graphs, ensuring the correct match based on orientation and location.