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
![**Determining the Equation of a Graphed Circle**
To find the equation of the circle depicted on the coordinate plane, follow these detailed steps:
### 1. Analyzing the Graph:
The graph presents a circle centered at the origin with the following characteristics:
- The horizontal axis (x-axis) runs from -10 to 10.
- The vertical axis (y-axis) runs from -10 to 10.
- The circle is completely contained within the grid.
### 2. Identify Key Components:
When determining the equation of a circle, we use the standard form of the circle's equation:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Where (h, k) is the center of the circle, and r is the radius.
### 3. Determine the Circle's Center:
From the graph, observe that the center of the circle is at the point (-6, 2).
### 4. Determine the Radius:
To find the radius, measure the distance from the center of the circle to any point on the circumference. By counting the grid squares from the center (-6, 2) to the edge of the circle, we see the radius is 2 units.
### 5. Formulate the Equation:
Substitute the center and radius values into the standard form:
- Center (h, k) = (-6, 2)
- Radius (r) = 2
\[ (x + 6)^2 + (y - 2)^2 = 4 \]
(Remember, \( r^2 = 2^2 = 4 \))
### Final Equation:
\[ (x + 6)^2 + (y - 2)^2 = 4 \]
This accurately represents the equation of the circle in the graph provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe98d4661-0645-4947-a693-dc215b41eab7%2F60fb7328-d129-4a5f-9031-05079e7c7abf%2Fuc140dp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determining the Equation of a Graphed Circle**
To find the equation of the circle depicted on the coordinate plane, follow these detailed steps:
### 1. Analyzing the Graph:
The graph presents a circle centered at the origin with the following characteristics:
- The horizontal axis (x-axis) runs from -10 to 10.
- The vertical axis (y-axis) runs from -10 to 10.
- The circle is completely contained within the grid.
### 2. Identify Key Components:
When determining the equation of a circle, we use the standard form of the circle's equation:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Where (h, k) is the center of the circle, and r is the radius.
### 3. Determine the Circle's Center:
From the graph, observe that the center of the circle is at the point (-6, 2).
### 4. Determine the Radius:
To find the radius, measure the distance from the center of the circle to any point on the circumference. By counting the grid squares from the center (-6, 2) to the edge of the circle, we see the radius is 2 units.
### 5. Formulate the Equation:
Substitute the center and radius values into the standard form:
- Center (h, k) = (-6, 2)
- Radius (r) = 2
\[ (x + 6)^2 + (y - 2)^2 = 4 \]
(Remember, \( r^2 = 2^2 = 4 \))
### Final Equation:
\[ (x + 6)^2 + (y - 2)^2 = 4 \]
This accurately represents the equation of the circle in the graph provided.
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