Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 44E
Related questions
Question
![### Graphing Parametric Equations and Rectangular Equations
**Problem Statement:**
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of the curve.
Parametric equations:
\[ x = t + 9, \quad y = \sqrt{t} \, ; \quad t \geq 0 \]
**Graph Selection:**
- Choose the correct graph from the options below:
- **Option A:**
A diagonal line starting from the point \((9, 0)\) and moving upwards to the right in the first quadrant.
- **Option B:**
A U-shaped curve that is open downwards.
- **Option C:**
An S-shaped curve that alternates in orientation.
- **Option D:**
A U-shaped curve that is open upwards in the positive direction of the y-axis and starting at \( (9,0) \)
**Rectangular Equation:**
The rectangular equation of the curve is calculated from the parametric equations.
**Solution Steps:**
1. Start with the given parametric equations:
\[ x = t + 9 \]
\[ y = \sqrt{t} \]
2. Solve the equation \( x = t + 9 \) for \( t \):
\[ t = x - 9 \]
3. Substitute \( t = x - 9 \) into the equation \( y = \sqrt{t} \):
\[ y = \sqrt{x - 9} \]
4. Therefore, the rectangular equation of the curve is:
\[ y = \sqrt{x - 9} \]
The correct graph should be selected from the options provided above which satisfies this rectangular equation.
**Final Answer:**
The rectangular equation of the curve is:
\[ y = \sqrt{x - 9} \]
Based on the graphs, Option **A** represents the correct graph of the rectangular equation:
\[ y = \sqrt{x - 9} \]
### Graph Explanation:
**Option A:**
- The graph correctly represents the equation \( y = \sqrt{x - 9}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe4962ae7-1cb2-4276-8f6a-b9f851b67289%2F9c1d221f-7868-4d59-9686-c282635e910f%2Fn7xlorg_processed.png&w=3840&q=75)
Transcribed Image Text:### Graphing Parametric Equations and Rectangular Equations
**Problem Statement:**
Graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of the curve.
Parametric equations:
\[ x = t + 9, \quad y = \sqrt{t} \, ; \quad t \geq 0 \]
**Graph Selection:**
- Choose the correct graph from the options below:
- **Option A:**
A diagonal line starting from the point \((9, 0)\) and moving upwards to the right in the first quadrant.
- **Option B:**
A U-shaped curve that is open downwards.
- **Option C:**
An S-shaped curve that alternates in orientation.
- **Option D:**
A U-shaped curve that is open upwards in the positive direction of the y-axis and starting at \( (9,0) \)
**Rectangular Equation:**
The rectangular equation of the curve is calculated from the parametric equations.
**Solution Steps:**
1. Start with the given parametric equations:
\[ x = t + 9 \]
\[ y = \sqrt{t} \]
2. Solve the equation \( x = t + 9 \) for \( t \):
\[ t = x - 9 \]
3. Substitute \( t = x - 9 \) into the equation \( y = \sqrt{t} \):
\[ y = \sqrt{x - 9} \]
4. Therefore, the rectangular equation of the curve is:
\[ y = \sqrt{x - 9} \]
The correct graph should be selected from the options provided above which satisfies this rectangular equation.
**Final Answer:**
The rectangular equation of the curve is:
\[ y = \sqrt{x - 9} \]
Based on the graphs, Option **A** represents the correct graph of the rectangular equation:
\[ y = \sqrt{x - 9} \]
### Graph Explanation:
**Option A:**
- The graph correctly represents the equation \( y = \sqrt{x - 9}
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