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Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 7 - Conic SectionsProblema 37
Capitolo 8, Problema 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

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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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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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Foci 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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Introduction to Asymptotes
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