Textbook Question
Graph each ellipse and locate the foci. x2/(9/4) +y2/(25/4) = 1
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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 7
Find the focus and directrix of the parabola with the given equation. Then graph the parabola. y2 = - 8x
Verified step by step guidance1
Identify the form of the given parabola equation. The equation is \(y^2 = -8x\), which matches the standard form of a horizontal parabola: \(y^2 = 4px\).
Compare the given equation \(y^2 = -8x\) with the standard form \(y^2 = 4px\) to find the value of \(p\). Here, \(4p = -8\), so solve for \(p\) by dividing both sides by 4.
Interpret the value of \(p\). Since \(p\) is negative, the parabola opens to the left. The vertex of the parabola is at the origin \((0,0)\) because the equation is not shifted.
Find the focus using the vertex and \(p\). The focus lies at \((p, 0)\), so substitute the value of \(p\) found in step 2 to get the coordinates of the focus.
Find the directrix. The directrix is a vertical line given by \(x = -p\). 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's equation helps identify its orientation and key features. For parabolas that open left or right, the form is y² = 4px, where p represents the distance from the vertex to the focus. Recognizing this form allows you to extract important information like the 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 used to define it, and the directrix is a line perpendicular to the axis of symmetry. The parabola consists of all points equidistant from the focus and directrix. Knowing how to calculate the focus and directrix from the equation is essential for graphing.
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Graphing Parabolas
Graphing a parabola involves plotting the vertex, focus, and directrix, then sketching the curve that is symmetric about the axis of symmetry. Understanding the direction the parabola opens (left, right, up, or down) and the distance p helps in accurately drawing the shape.
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Related Practice
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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Textbook Question
Find the standard form of the equation of each hyperbola satisfying the given conditions. Foci: (−4, 0), (4, 0); vertices:(−3, 0), (3, 0)
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Textbook Question
Find the focus and directrix of the parabola with the given equation. Then graph the parabola.
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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
Graph each ellipse and locate the foci. x2/49 +y2/81 = 1
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