To determine the equation of the circle from the given image, we need two key pieces of information: the center $C(h,k)$ and the radius $r$.

Step 1: Identify the center of the circle

From the graph, it is clear that the center $C$ of the circle is at the point $(0, 2)$.

Step 2: Calculate the radius

The radius $r$ is the distance between the center $C(0,2)$ and the point $P(4, 2)$ on the circle. The distance formula is:

$r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Using $C(0, 2)$ and $P(4, 2)$:

$r = \sqrt{(4 - 0)^2 + (2 - 2)^2} = \sqrt{16 + 0} = 4$

Thus, the radius $r = 4$.

Step 3: Write the equation of the circle

The general equation of a circle is:

$(x - h)^2 + (y - k)^2 = r^2$

Substituting $h = 0$, $k = 2$, and $r = 4$:

$(x - 0)^2 + (y - 2)^2 = 4^2$ $x^2 + (y - 2)^2 = 16$

Step 4: Select the correct option

The correct equation matches option C: $x^2 + (y - 2)^2 = 16$.

Would you like more details or clarifications? Here are 5 related questions to expand on this:

1. How would you find the equation of a circle if the center is not at the origin?
2. Can you describe the geometric meaning of each term in the circle equation?
3. What is the relationship between the radius and the circumference of a circle?
4. How does shifting the center of the circle affect the equation?
5. Can a circle have a negative radius in real-world geometry?

Tip: The radius is always the square root of the constant on the right side of the circle's equation.