Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Graphing Circles Example**
This exercise involves graphing the equation of a circle. The equation given is:
\[ (x - 1)^2 + (y - 3)^2 = 9 \]
The options provided show two graphs to help identify which one correctly represents this equation.
**Graph Descriptions:**
1. **First Graph**
- **X-axis and Y-axis**: Clearly labeled with increments of 1.
- **Grid**: Each square probably represents a single unit.
- **Circle Center**: Appears to be located at the point (1, 3).
- **Radius**: From the center, the circle extends 3 units in every direction (indicating a radius of 3).
2. **Second Graph**
- **X-axis and Y-axis**: Clearly labeled with increments of 1.
- **Grid**: Each square likely represents a single unit.
- **Circle Center**: Appears to be at the origin (0, 0).
- **Radius**: From the center, the circle extends 3 units in every direction, indicating the same radius of 3 for comparison.
**Explanation:**
For the equation \((x - 1)^2 + (y - 3)^2 = 9\), the center of the circle is at (1, 3) and the radius is \( \sqrt{9} = 3 \).
Therefore, the correct graph is the first one, as it accurately places the center of the circle at (1, 3) with a radius of 3 units.
**Conclusion:**
Understanding how to graph the equation of a circle involves recognizing the center and calculating the radius based on the standard form \((x - h)^2 + (y - k)^2 = r^2\). In this example, the correct graph is identified by checking these attributes against the given options. The first graph faithfully represents the given equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3b617bc-0d34-47a7-bf50-98a510ba10e9%2Fa9de4cb4-9bc0-4964-8129-64147b0f7a7f%2Fy7qa5b7_processed.png&w=3840&q=75)
Transcribed Image Text:**Graphing Circles Example**
This exercise involves graphing the equation of a circle. The equation given is:
\[ (x - 1)^2 + (y - 3)^2 = 9 \]
The options provided show two graphs to help identify which one correctly represents this equation.
**Graph Descriptions:**
1. **First Graph**
- **X-axis and Y-axis**: Clearly labeled with increments of 1.
- **Grid**: Each square probably represents a single unit.
- **Circle Center**: Appears to be located at the point (1, 3).
- **Radius**: From the center, the circle extends 3 units in every direction (indicating a radius of 3).
2. **Second Graph**
- **X-axis and Y-axis**: Clearly labeled with increments of 1.
- **Grid**: Each square likely represents a single unit.
- **Circle Center**: Appears to be at the origin (0, 0).
- **Radius**: From the center, the circle extends 3 units in every direction, indicating the same radius of 3 for comparison.
**Explanation:**
For the equation \((x - 1)^2 + (y - 3)^2 = 9\), the center of the circle is at (1, 3) and the radius is \( \sqrt{9} = 3 \).
Therefore, the correct graph is the first one, as it accurately places the center of the circle at (1, 3) with a radius of 3 units.
**Conclusion:**
Understanding how to graph the equation of a circle involves recognizing the center and calculating the radius based on the standard form \((x - h)^2 + (y - k)^2 = r^2\). In this example, the correct graph is identified by checking these attributes against the given options. The first graph faithfully represents the given equation.
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