Textbook Question
Find the standard form of the equation of each hyperbola satisfying the given conditions. Endpoints of transverse axis: (0, −6), (0, 6); asymptote: y=2x
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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 11
Find the focus and directrix of the parabola with the given equation. Then graph the parabola. x2 = - 16y
Verified step by step guidance1
Rewrite the given equation \(x^2 = -16y\) in the standard form of a vertical parabola, which is \(x^2 = 4py\). Identify the value of \$4p$ by comparing the two equations.
From the equation \(x^2 = 4py\), solve for \(p\) by dividing both sides by 4, so \(p = \frac{-16}{4}\). This value of \(p\) represents the distance from the vertex to the focus and from the vertex to the directrix.
Determine the vertex of the parabola. Since the equation is in the form \(x^2 = 4py\), the vertex is at the origin \((0,0)\).
Find the focus of the parabola. For a vertical parabola \(x^2 = 4py\), the focus is located at \((0, p)\). Use the value of \(p\) found in step 2 to write the coordinates of the focus.
Find the equation of the directrix. The directrix is a horizontal line given by \(y = -p\). Use the value of \(p\) to write the equation of the directrix.
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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 that opens vertically is x² = 4py, where p represents the distance from the vertex to the focus. Recognizing this form helps identify the parabola's orientation and key features such as the vertex, focus, and directrix.
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Parabolas as Conic Sections
Focus and Directrix of a Parabola
The focus is a fixed point inside the parabola, and the directrix is a line perpendicular to the axis of symmetry. For x² = 4py, the focus is at (0, p) and the directrix is y = -p. These elements define the parabola's shape and are essential for graphing.
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Graphing Parabolas
Graphing involves plotting the vertex, focus, and directrix, then sketching the curve that is equidistant from the focus and directrix. Understanding the parabola's orientation and key points allows accurate representation of its shape on the coordinate plane.
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Related Practice
Textbook Question
Find the standard form of the equation of each hyperbola satisfying the given conditions. Center: (4, −2); Focus: (7, −2); vertex: (6, −2)
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Textbook Question
Find the focus and directrix of the parabola with the given equation. Then graph the parabola. x2 = 12y
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Textbook Question
Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x2/9−y2/25=1
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Textbook Question
Graph each ellipse and locate the foci. 25x²+4y² = 100
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Textbook Question
Graph each ellipse and locate the foci. x² = 1 – 4y²
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