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Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 7 - Conic SectionsProblem 31
Chapter 8, Problem 31

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. Graph of a parabola opening upward with vertex at (1,1) on a Cartesian coordinate grid.b. Graph of a parabola opening left with vertex at (1,1) on a Cartesian coordinate grid.c. Graph of a right-opening parabola with vertex at (1,1) plotted on an x-y coordinate grid.d. Graph of a downward-opening parabola with vertex at (1,1) on a coordinate plane with labeled axes.

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.
Determine the vertex of the parabola. The vertex is given by the point \((h, k)\), so here the vertex is at \((1, 1)\).
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.
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.
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Parabolas as Conic Sections

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.
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Vertex Form

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.
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Parabolas as Conic Sections
Related Practice
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