Textbook Question
Find the vertex, focus, and directrix of the parabola with the given equation. Then graph the parabola. x^2 - 4x - 2y = 0
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 8, Problem 41
Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. (y + 1)2 = - 8x
Verified step by step guidance1
Identify the form of the given equation. The equation \((y + 1)^2 = -8x\) is a parabola in the form \((y - k)^2 = 4p(x - h)\), which represents a parabola that opens either left or right.
Rewrite the equation to match the standard form: \((y - (-1))^2 = -8(x - 0)\). Here, the vertex is at \((h, k) = (0, -1)\).
Determine the value of \(p\) by comparing \$4p\( to the coefficient of \)(x - h)\(. Since \(4p = -8\), solve for \)p\( to get \)p = -2$. The negative sign indicates the parabola opens to the left.
Find the focus using the vertex and \(p\). The focus lies \(p\) units from the vertex along the axis of symmetry. Since the parabola opens left, the focus is at \((h + p, k) = (0 - 2, -1)\).
Find the directrix, which is a vertical line \(p\) units in the opposite direction from the vertex. The directrix is the line \(x = h - p = 0 - (-2) = 2\).
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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
A parabola can be expressed in standard form as (y - k)^2 = 4p(x - h) for horizontal parabolas, where (h, k) is the vertex. This form helps identify the vertex, direction, and width of the parabola. Recognizing and rewriting the equation into this form is essential for further analysis.
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Parabolas as Conic Sections
Vertex, Focus, and Directrix of a Parabola
The vertex is the parabola's turning point, the focus is a fixed point inside the curve, and the directrix is a line outside the curve. The distance from the vertex to the focus and directrix is |p|, where p determines the parabola's shape and orientation. These elements define the parabola's geometry.
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7:18Horizontal Parabolas Example 1
Graphing Parabolas
Graphing involves plotting the vertex, focus, and directrix, then sketching the curve symmetric about the axis through the vertex and focus. Understanding the parabola's orientation (opening left, right, up, or down) and scale based on |4p| helps create an accurate graph.
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5:28Horizontal Parabolas
Related Practice
Textbook Question
Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. x2 - 2x - 4y + 9 =0
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Textbook Question
Find the standard form of the equation of the parabola satisfying the given conditions. Focus: (12,0); Directrix: x=-12
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Textbook Question
Find the standard form of the equation of the parabola satisfying the given conditions. Focus: (0,-11); Directrix: y=11
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Textbook Question
Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x−1)2−(y−2)2=3
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Textbook Question
Graph each ellipse and give the location of its foci. (x − 4)²/9 + (y +2)² /25= 1
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