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Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 7 - Conic SectionsProblem 33
Chapter 8, Problem 33

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x+4)2/9−(y+3)2/16=1

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Identify the center of the hyperbola from the equation. The equation is given as \(\frac{(x+4)^2}{9} - \frac{(y+3)^2}{16} = 1\), so the center is at \((-4, -3)\).
Determine the values of \(a^2\) and \(b^2\) from the denominators. Here, \(a^2 = 9\) and \(b^2 = 16\), so \(a = 3\) and \(b = 4\). Since the \(x\)-term is positive and comes first, the hyperbola opens left and right (horizontal transverse axis).
Locate the vertices using the center and \(a\). The vertices are \(a\) units left and right from the center along the \(x\)-axis, so the vertices are at \((-4 - 3, -3)\) and \((-4 + 3, -3)\).
Find the foci using the relationship \(c^2 = a^2 + b^2\). Calculate \(c\) by \(c = \sqrt{9 + 16}\), then locate the foci \(c\) units left and right from the center along the \(x\)-axis, at \((-4 - c, -3)\) and \((-4 + c, -3)\).
Write the equations of the asymptotes. For a hyperbola with a horizontal transverse axis, the asymptotes are given by \(y - k = \pm \frac{b}{a}(x - h)\), where \((h, k)\) is the center. Substitute \(h = -4\), \(k = -3\), \(a = 3\), and \(b = 4\) 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 written as (x-h)^2/a^2 - (y-k)^2/b^2 = 1 or (y-k)^2/a^2 - (x-h)^2/b^2 = 1, where (h, k) is the center. The sign in front of each term determines the hyperbola's orientation (horizontal or vertical). Understanding this form helps identify the center, vertices, and the relationship between a and b.
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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 are points further from the center, located c units away, where c^2 = a^2 + b^2. These points are essential for graphing and understanding the hyperbola's shape and properties.
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Foci and Vertices of Hyperbolas

Equations of Asymptotes

Asymptotes are lines that the hyperbola approaches but never touches, guiding its shape. For a hyperbola centered at (h, k), the asymptotes' equations are y - k = ±(b/a)(x - h) for horizontal transverse axis, or y - k = ±(a/b)(x - h) for vertical. They help in sketching the hyperbola accurately.
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Related Practice
Textbook Question

Find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). (x + 1)2 = - 4(y + 1)

a. b. c. d.

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Textbook Question

Find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). (y - 1)2 = 4(x - 1)

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In Exercises 31–34, find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). (y - 1)2 = - 4(x - 1)


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Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis vertical with length 10; length of minor axis = 4; center: (-2, 3)

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Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x+3)2/25−y2/16=1

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