Textbook Question
Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. (x + 1)2 = - 8(y + 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 37
Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (y+2)2/4−(x−1)2/16=1
Verified step by step guidance1
Identify the center of the hyperbola from the equation \(\frac{(y+2)^2}{4} - \frac{(x-1)^2}{16} = 1\). The center is at the point \((h, k) = (1, -2)\).
Determine the values of \(a^2\) and \(b^2\) from the denominators. Here, \(a^2 = 4\) and \(b^2 = 16\). Since the \(y\)-term is positive and comes first, the hyperbola opens vertically.
Find the vertices by moving \(a\) units up and down from the center along the \(y\)-axis. The vertices are at \((1, -2 \pm a)\), where \(a = \sqrt{4}\).
Calculate the foci using the relationship \(c^2 = a^2 + b^2\). Find \(c\) and then locate the foci at \((1, -2 \pm c)\) along the vertical axis.
Write the equations of the asymptotes using the formula for vertical hyperbolas: \(y = k \pm \frac{a}{b}(x - h)\). Substitute \(a\), \(b\), \(h\), and \(k\) to get the asymptote equations.
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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 (y-k)^2/a^2 - (x-h)^2/b^2 = 1 or (x-h)^2/a^2 - (y-k)^2/b^2 = 1, where (h, k) is the center. The sign and position of terms determine the hyperbola's orientation (vertical or horizontal). This form helps identify key features like vertices, foci, and asymptotes.
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Asymptotes of Hyperbolas
Vertices and Foci of a Hyperbola
Vertices are points on the hyperbola closest to the center, located a units away along the transverse axis. Foci lie further out, at a distance c from the center, where c^2 = a^2 + b^2. Knowing vertices and foci is essential for graphing and understanding the hyperbola's shape.
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5:22Foci and Vertices of Hyperbolas
Equations of Asymptotes
Asymptotes are lines that the hyperbola approaches but never touches, passing through the center. For a vertical hyperbola, asymptotes have equations y - k = ±(a/b)(x - h). These lines guide the hyperbola's shape and are crucial for accurate graphing.
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6:24Introduction to Asymptotes
Related Practice
Textbook Question
Find the vertex, focus, and directrix of the parabola with the given equation. Then graph the parabola. (y-2)^2 = -16x
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Textbook Question
Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. (x - 2)2 = 8(y - 1)
Textbook Question
Graph each ellipse and give the location of its foci. (x − 2)²/9 + (y -1)² /4= 1
Textbook Question
Graph each ellipse and give the location of its foci. (x +3)²+ 4(y -2)² = 16
Textbook Question
Find the vertex, focus, and directrix of the parabola with the given equation. Then graph the parabola. y^2 = 8x