Which of the following is a parameterization of the line that passes through the point (-1, -2) with a slope of 4?
Which of the following is a parameterization of the line that passes through the point (-1, -2) with a slope of 4?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 94E
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![### Parameterization of a Line through a Given Point with a Specific Slope
**Question:**
Which of the following is a parameterization of the line that passes through the point \((-1, -2)\) with a slope of 4?
**Options:**
1. \( x = 3t \) and \( y = 4t - 2 \), for any \( t \)
2. \( x = t + 3 \) and \( y = 4t + 1 \), for any \( t \)
3. \( x = 2t \) and \( y = 4t - 7 \), for any \( t \)
4. \( x = \tan t \) and \( y = 4\tan t + 2 \), for \( -\frac{\pi}{2} < t < \frac{\pi}{2} \)
**Analysis:**
- A parameterization of a line in a plane involves expressing the coordinates \( (x, y) \) as functions of a parameter \( t \).
- The general form of the parameterization of a line passing through a point \( (x_0, y_0) \) with slope \( m \) is:
\[ x = x_0 + at \]
\[ y = y_0 + bt \]
where \( b = ma \).
- Given the point \( (-1, -2) \) and slope \( 4 \), the correct parameterization must reflect these characteristics.
- Let's analyze each option to see which fits the requirements.
**Option 1:**
\[ x = 3t \]
\[ y = 4t - 2 \]
- This does not pass through \( (-1, -2) \) when \( t = 0 \).
**Option 2:**
\[ x = t + 3 \]
\[ y = 4t + 1 \]
- This does not pass through \( (-1, -2) \) when \( t = 0 \).
**Option 3:**
\[ x = 2t \]
\[ y = 4t - 7 \]
- This does not pass through \( (-1, -2) \) when \( t = 0 \).
**Option 4:**
\[ x = \tan t \]
\[ y = 4 \tan t + 2 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F361784a8-d964-492d-9687-5d5c5e192aef%2F19399f32-f9f3-4980-bc8d-f5cae3d0740d%2Fa8wpdv_processed.png&w=3840&q=75)
Transcribed Image Text:### Parameterization of a Line through a Given Point with a Specific Slope
**Question:**
Which of the following is a parameterization of the line that passes through the point \((-1, -2)\) with a slope of 4?
**Options:**
1. \( x = 3t \) and \( y = 4t - 2 \), for any \( t \)
2. \( x = t + 3 \) and \( y = 4t + 1 \), for any \( t \)
3. \( x = 2t \) and \( y = 4t - 7 \), for any \( t \)
4. \( x = \tan t \) and \( y = 4\tan t + 2 \), for \( -\frac{\pi}{2} < t < \frac{\pi}{2} \)
**Analysis:**
- A parameterization of a line in a plane involves expressing the coordinates \( (x, y) \) as functions of a parameter \( t \).
- The general form of the parameterization of a line passing through a point \( (x_0, y_0) \) with slope \( m \) is:
\[ x = x_0 + at \]
\[ y = y_0 + bt \]
where \( b = ma \).
- Given the point \( (-1, -2) \) and slope \( 4 \), the correct parameterization must reflect these characteristics.
- Let's analyze each option to see which fits the requirements.
**Option 1:**
\[ x = 3t \]
\[ y = 4t - 2 \]
- This does not pass through \( (-1, -2) \) when \( t = 0 \).
**Option 2:**
\[ x = t + 3 \]
\[ y = 4t + 1 \]
- This does not pass through \( (-1, -2) \) when \( t = 0 \).
**Option 3:**
\[ x = 2t \]
\[ y = 4t - 7 \]
- This does not pass through \( (-1, -2) \) when \( t = 0 \).
**Option 4:**
\[ x = \tan t \]
\[ y = 4 \tan t + 2 \
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