Textbook Question
Find the focus and directrix of the parabola with the given equation. Then graph the parabola. 8x2 + 4y = 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 13
Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x2/9−y2/25=1
Verified step by step guidance1
Identify the standard form of the hyperbola equation given: \(\frac{x^{2}}{9} - \frac{y^{2}}{25} = 1\). This is a hyperbola centered at the origin with the \(x^{2}\) term positive, indicating it opens left and right.
Determine the values of \(a^{2}\) and \(b^{2}\) from the denominators: \(a^{2} = 9\) and \(b^{2} = 25\). Then find \(a = \sqrt{9} = 3\) and \(b = \sqrt{25} = 5\).
Locate the vertices on the \(x\)-axis at \((\pm a, 0)\), which are \((\pm 3, 0)\), since the hyperbola opens horizontally.
Find the foci using the relationship \(c^{2} = a^{2} + b^{2}\). Calculate \(c = \sqrt{9 + 25}\), then place the foci at \((\pm c, 0)\) along the \(x\)-axis.
Write the equations of the asymptotes for a hyperbola centered at the origin with horizontal transverse axis: \(y = \pm \frac{b}{a} x\). Substitute \(a\) and \(b\) to get \(y = \pm \frac{5}{3} x\).
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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 Hyperbola
A hyperbola's equation in standard form is either (x²/a²) - (y²/b²) = 1 or (y²/a²) - (x²/b²) = 1. This form helps identify the orientation of the hyperbola (horizontal or vertical), the lengths of the transverse and conjugate axes, and the locations of vertices and asymptotes.
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Asymptotes of Hyperbolas
Vertices and Foci of a Hyperbola
Vertices are points where the hyperbola intersects its transverse axis, located at a distance 'a' from the center. Foci lie further along this axis at a distance 'c', where c² = a² + b². Knowing vertices and foci is essential for graphing and understanding the hyperbola's shape.
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Equations of Asymptotes for a Hyperbola
Asymptotes are lines that the hyperbola approaches but never touches. For a hyperbola centered at the origin, the asymptotes have equations y = ±(b/a)x for horizontal hyperbolas and y = ±(a/b)x for vertical ones. These lines guide the graph's shape and help in sketching the hyperbola.
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5:50Asymptotes of Hyperbolas
Related Practice
Textbook Question
In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x2/100−y2/64=1
Textbook Question
Graph the hyperbola. Locate the foci and find the equations of the asymptotes. (x^2)/16 - y^2 = 1
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Textbook Question
Find the focus and directrix of the parabola with the given equation. Then graph the parabola. x2 = - 16y
Textbook Question
Find the focus and directrix of the parabola with the given equation. Then graph the parabola. y2 - 6x = 0
Textbook Question
Graph each ellipse and locate the foci. 25x²+4y² = 100