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Use the center, vertices, and asymptotes to graph the hyperbola. Locate the foci and find the equations of the asymptotes. (x+2)² (y+3)² 9 25 = 1 Graph the hyperbola. Choose the correct graph below. O A. O B. O C. D. 1

Use the center, vertices, and asymptotes to graph the hyperbola. Locate the foci and find the equations of the asymptotes. (x+2)² (y+3)² 9 25 = 1 Graph the hyperbola. Choose the correct graph below. O A. O B. O C. D. 1

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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To graph the hyperbola and find its key components, use the center, vertices, and asymptotes. The given hyperbola equation is: \[ \frac{(x + 2)^2}{9} - \frac{(y + 3)^2}{25} = 1 \] ### Steps to Follow: 1. **Identify the Center:** - The center of the hyperbola is at \((-2, -3)\). 2. **Vertices:** - The transverse axis is horizontal, as the \((x + 2)^2\) term is positive. - The distance to each vertex from the center is given by \(\sqrt{9} = 3\). - Thus, the vertices are at: \((-2 + 3, -3) = (1, -3)\) and \((-2 - 3, -3) = (-5, -3)\). 3. **Asymptotes:** - The slopes of the asymptotes for a hyperbola centered at \((h, k)\) with horizontal transverse axis are given by \(\pm \frac{b}{a}\). - Here, \(a = \sqrt{9} = 3\) and \(b = \sqrt{25} = 5\). - The equations of the asymptotes are: - \(y + 3 = \frac{5}{3}(x + 2)\) - \(y + 3 = -\frac{5}{3}(x + 2)\) 4. **Foci:** - Calculate the foci using the formula \(c = \sqrt{a^2 + b^2}\). - \(c = \sqrt{9 + 25} = \sqrt{34}\). - The foci are located at: \((-2 + \sqrt{34}, -3)\) and \((-2 - \sqrt{34}, -3)\). ### Choose the Correct Graph: Refer to the four options labeled A, B, C, D: - **Graph A:** Shows a hyperbola with the horizontal transverse axis, vertices approximately at \((-5, -3)\) and \((1, -3)\), and asymptotes \(\pm \frac{5}{3}\). - **Graph B, C, D:** Do not match the
Transcribed Image Text:To graph the hyperbola and find its key components, use the center, vertices, and asymptotes. The given hyperbola equation is: \[ \frac{(x + 2)^2}{9} - \frac{(y + 3)^2}{25} = 1 \] ### Steps to Follow: 1. **Identify the Center:** - The center of the hyperbola is at \((-2, -3)\). 2. **Vertices:** - The transverse axis is horizontal, as the \((x + 2)^2\) term is positive. - The distance to each vertex from the center is given by \(\sqrt{9} = 3\). - Thus, the vertices are at: \((-2 + 3, -3) = (1, -3)\) and \((-2 - 3, -3) = (-5, -3)\). 3. **Asymptotes:** - The slopes of the asymptotes for a hyperbola centered at \((h, k)\) with horizontal transverse axis are given by \(\pm \frac{b}{a}\). - Here, \(a = \sqrt{9} = 3\) and \(b = \sqrt{25} = 5\). - The equations of the asymptotes are: - \(y + 3 = \frac{5}{3}(x + 2)\) - \(y + 3 = -\frac{5}{3}(x + 2)\) 4. **Foci:** - Calculate the foci using the formula \(c = \sqrt{a^2 + b^2}\). - \(c = \sqrt{9 + 25} = \sqrt{34}\). - The foci are located at: \((-2 + \sqrt{34}, -3)\) and \((-2 - \sqrt{34}, -3)\). ### Choose the Correct Graph: Refer to the four options labeled A, B, C, D: - **Graph A:** Shows a hyperbola with the horizontal transverse axis, vertices approximately at \((-5, -3)\) and \((1, -3)\), and asymptotes \(\pm \frac{5}{3}\). - **Graph B, C, D:** Do not match the
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