2. Find the vertex, focus, and directrix of the parabola. Then sketch the graph. (x − 3)² = 8(y + 1)
2. Find the vertex, focus, and directrix of the parabola. Then sketch the graph. (x − 3)² = 8(y + 1)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 2**: Find the vertex, focus, and directrix of the parabola. Then sketch the graph.
Given equation:
\[
(x - 3)^2 = 8(y + 1)
\]
### Solution
To find the vertex, focus, and directrix of the parabola, we start with the given equation:
\[
(x - 3)^2 = 8(y + 1)
\]
This is in the form \((x - h)^2 = 4p(y - k)\), where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the focus and directrix.
**1. Vertex**:
The vertex \((h, k)\) is \((3, -1)\).
**2. Focus**:
Since \(4p = 8\), we have \(p = 2\). The focus is located at \((h, k + p)\). Therefore, the focus is at \((3, 1)\).
**3. Directrix**:
The directrix is the line \(y = k - p\). So, the directrix is \(y = -3\).
### Graph Explanation
- The vertex of the parabola is at the point \((3, -1)\).
- The parabola opens upwards because the coefficient of \((y + 1)\) is positive.
- The focus \((3, 1)\) is inside the parabola, 2 units above the vertex.
- The directrix \(y = -3\) is a horizontal line below the vertex, 2 units below it.
The sketch of the parabola will be symmetric about the vertical line \(x = 3\) and will pass through the focus and be equidistant from the directrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87eb72d4-41bd-40a9-b4c9-a26630078586%2F8d37e6a6-d53d-4959-88fa-e1bd1bedf2c1%2F1y8zz1d_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 2**: Find the vertex, focus, and directrix of the parabola. Then sketch the graph.
Given equation:
\[
(x - 3)^2 = 8(y + 1)
\]
### Solution
To find the vertex, focus, and directrix of the parabola, we start with the given equation:
\[
(x - 3)^2 = 8(y + 1)
\]
This is in the form \((x - h)^2 = 4p(y - k)\), where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the focus and directrix.
**1. Vertex**:
The vertex \((h, k)\) is \((3, -1)\).
**2. Focus**:
Since \(4p = 8\), we have \(p = 2\). The focus is located at \((h, k + p)\). Therefore, the focus is at \((3, 1)\).
**3. Directrix**:
The directrix is the line \(y = k - p\). So, the directrix is \(y = -3\).
### Graph Explanation
- The vertex of the parabola is at the point \((3, -1)\).
- The parabola opens upwards because the coefficient of \((y + 1)\) is positive.
- The focus \((3, 1)\) is inside the parabola, 2 units above the vertex.
- The directrix \(y = -3\) is a horizontal line below the vertex, 2 units below it.
The sketch of the parabola will be symmetric about the vertical line \(x = 3\) and will pass through the focus and be equidistant from the directrix.
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