Use the function (x-3)2 + y2 = 4 to answer the following questions. Is the function an equation of a circle and why

Let's analyze the function \((x - 3)^2 + y^2 = 4\) to determine if it represents the equation of a circle.

### Standard Equation of a Circle
The standard form of a circle's equation is:

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

where:
- \((h, k)\) is the center of the circle
- \(r\) is the radius of the circle

### Analyzing the Given Function
The given function is:

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

This matches the standard form, where:
- \(h = 3\) (the x-coordinate of the center)
- \(k = 0\) (the y-coordinate of the center, since \(y^2\) is the same as \((y - 0)^2\))
- \(r^2 = 4\), so \(r = 2\)

### Conclusion
Yes, the function \((x - 3)^2 + y^2 = 4\) is indeed the equation of a circle. The reason is that it fits the standard form of a circle's equation, with the center at \((3, 0)\) and a radius of 2.

Would you like more details or have any questions?

Here are 8 related questions:
1. What are the coordinates of the center for the given circle equation?
2. How do you find the radius from the circle equation?
3. How can you graph the circle based on this equation?
4. How would the equation change if the circle's center were at the origin?
5. Can this equation be converted to a general form?
6. What happens if the radius is zero?
7. How do you determine if another point lies on this circle?
8. How does the equation change if you translate the circle along the y-axis?

**Tip:** When analyzing an equation to determine if it's a circle, always compare it to the standard form \((x - h)^2 + (y - k)^2 = r^2\) to identify the center and radius.

Explore the equation (x-3)^2 + y^2 = 4 and learn how it defines a circle. This detailed analysis covers finding the center, determining the radius, and graphing the circle. Suitable for Grades 7-9.

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