Use the function (x-3)2 + y2 = 4 to answer the following questions

a. Is the function an equation of a circle? 

b. If yes. What are the coordinates and radius of the circle?

To determine whether the given function is the equation of a circle and to find the coordinates and radius if it is, we need to analyze the given equation:

\[ (x - 3)^2 + y^2 = 4 \]

### a. Is the function an equation of a circle?

Yes, the given function is in the standard form of a circle's equation, which is:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

where \((h, k)\) are the coordinates of the center of the circle and \(r\) is the radius.

### b. What are the coordinates and radius of the circle?

To identify the center and radius, we compare the given equation to the standard form:

\[ (x - 3)^2 + y^2 = 4 \]

From this comparison:
- \((h, k)\) is \((3, 0)\)
- \(r^2\) is \(4\), so \(r = \sqrt{4} = 2\)

Thus:
- The coordinates of the center of the circle are \((3, 0)\).
- The radius of the circle is \(2\).

Would you like more details or have any questions?

Here are some related questions:

1. How do you derive the standard form of the equation of a circle from its general form?
2. What is the general form of the equation of a circle?
3. How can you determine if a given point lies on the circle?
4. What is the process to find the intersection points of a circle and a line?
5. How do you complete the square to convert a quadratic equation to the standard form of a circle?
6. What are the key properties of a circle in the coordinate plane?
7. How do transformations affect the equation of a circle?
8. How do you find the tangent line to a circle at a given point?

**Tip:** When given an equation of the form \((x - h)^2 + (y - k)^2 = r^2\), remember that \((h, k)\) represents the center of the circle, and \(r\) represents the radius, derived from taking the square root of the constant on the right side.

Explore the equation (x-3)^2 + y^2 = 4, determine if it represents a circle, and learn how to find its center and radius. This problem covers key concepts in coordinate geometry and circle equations. Suitable for Grades 9-12 students.

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