Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Drawing a Circle from an Equation
**Task:** Draw a circle with an equation of \( x^2 - 2x + y^2 + 2y = 7 \).
#### Steps to Solve:
1. **Rewrite the Equation:**
- Start by completing the square for both the \( x \) and \( y \) terms in the given equation.
\[
x^2 - 2x + y^2 + 2y = 7
\]
2. **Complete the Square:**
- For \( x \):
\[
x^2 - 2x = (x - 1)^2 - 1
\]
- For \( y \):
\[
y^2 + 2y = (y + 1)^2 - 1
\]
3. **Rewrite the Equation with Completed Squares:**
- Substitute back into the original equation:
\[
(x - 1)^2 - 1 + (y + 1)^2 - 1 = 7
\]
- Simplify:
\[
(x - 1)^2 + (y + 1)^2 - 2 = 7
\]
\[
(x - 1)^2 + (y + 1)^2 = 9
\]
4. **Identify the Circle’s Center and Radius:**
- The equation \((x - 1)^2 + (y + 1)^2 = 9\) represents a circle with center \((1, -1)\) and radius \(3\).
5. **Graph the Circle:**
- Plot the center of the circle at \((1, -1)\) on the coordinate plane.
- Use a compass or careful measurement to plot points that are exactly 3 units away from the center in all directions to form the circle.
#### Explanation of the Graph:
- **Axes and Quadrants:**
- The graph is a standard Cartesian coordinate plane spanning from -5 to 5 on both the x and y axes.
- Each axis is labeled with integers from -5 to 5.
- **Plotting the Center and Radius:**
- The center of the circle is at the coordinate \((1, -1)\), which is one unit to the right of the y-axis and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f072395-e0a4-486f-9c92-023b53ab9ed9%2F03b3890d-1ce5-4675-9238-2cb615bbc04a%2F950okpr_processed.png&w=3840&q=75)
Transcribed Image Text:### Drawing a Circle from an Equation
**Task:** Draw a circle with an equation of \( x^2 - 2x + y^2 + 2y = 7 \).
#### Steps to Solve:
1. **Rewrite the Equation:**
- Start by completing the square for both the \( x \) and \( y \) terms in the given equation.
\[
x^2 - 2x + y^2 + 2y = 7
\]
2. **Complete the Square:**
- For \( x \):
\[
x^2 - 2x = (x - 1)^2 - 1
\]
- For \( y \):
\[
y^2 + 2y = (y + 1)^2 - 1
\]
3. **Rewrite the Equation with Completed Squares:**
- Substitute back into the original equation:
\[
(x - 1)^2 - 1 + (y + 1)^2 - 1 = 7
\]
- Simplify:
\[
(x - 1)^2 + (y + 1)^2 - 2 = 7
\]
\[
(x - 1)^2 + (y + 1)^2 = 9
\]
4. **Identify the Circle’s Center and Radius:**
- The equation \((x - 1)^2 + (y + 1)^2 = 9\) represents a circle with center \((1, -1)\) and radius \(3\).
5. **Graph the Circle:**
- Plot the center of the circle at \((1, -1)\) on the coordinate plane.
- Use a compass or careful measurement to plot points that are exactly 3 units away from the center in all directions to form the circle.
#### Explanation of the Graph:
- **Axes and Quadrants:**
- The graph is a standard Cartesian coordinate plane spanning from -5 to 5 on both the x and y axes.
- Each axis is labeled with integers from -5 to 5.
- **Plotting the Center and Radius:**
- The center of the circle is at the coordinate \((1, -1)\), which is one unit to the right of the y-axis and
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