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4) vertices at (5, -2), (1, -2); asymptotes y = -2 ± - (x − 3) 5) center (8, -3); focus (8, -10); vertex at (8, -7)

4) vertices at (5, -2), (1, -2); asymptotes y = -2 ± - (x − 3) 5) center (8, -3); focus (8, -10); vertex at (8, -7)

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter12: Conic Sections
Section12.CR: Chapter Review
Problem 7CC
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Write an equation for each hyperbola
### Conic Sections: Hyperbolas and Ellipses **Example Problem 1: Hyperbola Analysis** Given the information: **Vertices** at (5, -2) and (1, -2) **Asymptotes**: \( y = -2 \pm \frac{3}{2}(x - 3) \) In this problem, the vertices are points where the hyperbola intersects its principal axis. The asymptotes provide the lines that the hyperbola approaches at infinity. **Example Problem 2: Ellipse Analysis** Given the information: **Center** at (8, -3) **Focus** at (8, -10) **Vertex** at (8, -7) For this ellipse, the center is the midpoint of the major and minor axes. The focus is a point used in the ellipse's definition, lying along its major axis. The vertex is where the ellipse intersects its principal axis (major axis). **Graph Explanation:** There is a small partial graph visible. It appears to be a segment of a standard coordinate grid, marked clearly with axis labels. Such graphs are typically used for plotting points, functions, or conic sections by hand. In this context, you could plot the given vertices, foci, and asymptotes: 1. **For the Hyperbola**: - Plot the vertices (5, -2) and (1, -2). - Sketch the asymptotes using the formula \( y = -2 \pm \frac{3}{2}(x - 3) \), which will form two lines intersecting at the hyperbola's center. 2. **For the Ellipse**: - Mark the center at (8, -3). - Plot the focus at (8, -10). - Plot the vertex at (8, -7), and use the standard ellipse formula \((x - h)^2 / a^2 + (y - k)^2 / b^2 = 1\) to relate these points. For educational purposes, these examples illustrate how to identify key features of conic sections, such as vertices, foci, and asymptotes for hyperbolas, and centers, foci, and vertices for ellipses.
Transcribed Image Text:### Conic Sections: Hyperbolas and Ellipses **Example Problem 1: Hyperbola Analysis** Given the information: **Vertices** at (5, -2) and (1, -2) **Asymptotes**: \( y = -2 \pm \frac{3}{2}(x - 3) \) In this problem, the vertices are points where the hyperbola intersects its principal axis. The asymptotes provide the lines that the hyperbola approaches at infinity. **Example Problem 2: Ellipse Analysis** Given the information: **Center** at (8, -3) **Focus** at (8, -10) **Vertex** at (8, -7) For this ellipse, the center is the midpoint of the major and minor axes. The focus is a point used in the ellipse's definition, lying along its major axis. The vertex is where the ellipse intersects its principal axis (major axis). **Graph Explanation:** There is a small partial graph visible. It appears to be a segment of a standard coordinate grid, marked clearly with axis labels. Such graphs are typically used for plotting points, functions, or conic sections by hand. In this context, you could plot the given vertices, foci, and asymptotes: 1. **For the Hyperbola**: - Plot the vertices (5, -2) and (1, -2). - Sketch the asymptotes using the formula \( y = -2 \pm \frac{3}{2}(x - 3) \), which will form two lines intersecting at the hyperbola's center. 2. **For the Ellipse**: - Mark the center at (8, -3). - Plot the focus at (8, -10). - Plot the vertex at (8, -7), and use the standard ellipse formula \((x - h)^2 / a^2 + (y - k)^2 / b^2 = 1\) to relate these points. For educational purposes, these examples illustrate how to identify key features of conic sections, such as vertices, foci, and asymptotes for hyperbolas, and centers, foci, and vertices for ellipses.
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