Question 8 On the coordinate plane below, graph the hyperbola represented by the following equation. = 1 16 . Draw the auxiliary rectangle and the asymptotes. ● Draw and label the vertices, the center, and the two focal points.
Question 8 On the coordinate plane below, graph the hyperbola represented by the following equation. = 1 16 . Draw the auxiliary rectangle and the asymptotes. ● Draw and label the vertices, the center, and the two focal points.
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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![**Question 8**
On the coordinate plane below, graph the hyperbola represented by the following equation.
\[
\frac{x^2}{9} - \frac{y^2}{16} = 1
\]
- Draw the auxiliary rectangle and the asymptotes.
- Draw and label the vertices, the center, and the two focal points.
The following is a description of the graph:
1. **Axes and Grid**: The graph is plotted on a rectangular coordinate system with the x-axis and y-axis intersecting at the origin (0,0). The grid has intervals indicating 1 unit per division on both axes.
2. **Hyperbola Equation**: The given equation is \(\frac{x^2}{9} - \frac{y^2}{16} = 1\), which is the standard form for a hyperbola centered at the origin with the transverse axis along the x-axis.
3. **Vertices and Center**:
- **Center (C)**: The center of the hyperbola is at the origin (0,0).
- **Vertices (V1 and V2)**: The vertices are located at (\(\pm a, 0\)), where \(a = \sqrt{9} = 3\). Thus, the vertices are at \(V1 (3, 0)\) and \(V2 (-3, 0)\).
4. **Focal Points**:
- The focal points (F1 and F2) lie along the transverse axis (x-axis) and can be found using the relationship \(c = \sqrt{a^2 + b^2}\), where \(c\) is the distance from the center to each focus.
- Here, \(a^2 = 9\) and \(b^2 = 16\).
- Thus, \(c = \sqrt{9 + 16} = \sqrt{25} = 5\).
- The focal points are located at \(F1 (5, 0)\) and \(F2 (-5, 0)\).
5. **Auxiliary Rectangle and Asymptotes**:
- The auxiliary rectangle helps in sketching the asymptotes.
- The sides of this rectangle are parallel to the axes and pass through the vertices and co-vertices. The dimensions are \(2a\) along the x-axis and \(2b\) along](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8182640d-a39f-4906-bb1d-1d29c4c23cc2%2F56bc1cbd-6502-4145-a50c-3ef73fe6ef8d%2Feslu8uq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 8**
On the coordinate plane below, graph the hyperbola represented by the following equation.
\[
\frac{x^2}{9} - \frac{y^2}{16} = 1
\]
- Draw the auxiliary rectangle and the asymptotes.
- Draw and label the vertices, the center, and the two focal points.
The following is a description of the graph:
1. **Axes and Grid**: The graph is plotted on a rectangular coordinate system with the x-axis and y-axis intersecting at the origin (0,0). The grid has intervals indicating 1 unit per division on both axes.
2. **Hyperbola Equation**: The given equation is \(\frac{x^2}{9} - \frac{y^2}{16} = 1\), which is the standard form for a hyperbola centered at the origin with the transverse axis along the x-axis.
3. **Vertices and Center**:
- **Center (C)**: The center of the hyperbola is at the origin (0,0).
- **Vertices (V1 and V2)**: The vertices are located at (\(\pm a, 0\)), where \(a = \sqrt{9} = 3\). Thus, the vertices are at \(V1 (3, 0)\) and \(V2 (-3, 0)\).
4. **Focal Points**:
- The focal points (F1 and F2) lie along the transverse axis (x-axis) and can be found using the relationship \(c = \sqrt{a^2 + b^2}\), where \(c\) is the distance from the center to each focus.
- Here, \(a^2 = 9\) and \(b^2 = 16\).
- Thus, \(c = \sqrt{9 + 16} = \sqrt{25} = 5\).
- The focal points are located at \(F1 (5, 0)\) and \(F2 (-5, 0)\).
5. **Auxiliary Rectangle and Asymptotes**:
- The auxiliary rectangle helps in sketching the asymptotes.
- The sides of this rectangle are parallel to the axes and pass through the vertices and co-vertices. The dimensions are \(2a\) along the x-axis and \(2b\) along
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