The equation of a circle can be derived using its general form:

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

where:

• $(h, k)$ is the center of the circle.
• $r$ is the radius.

From the graph you provided, we can see that the center of the circle is at $(-6, 4)$. To determine the radius, we need to calculate the distance between the center and any point on the circle. For example, we can use the point $(-3, 6)$, which lies on the circle.

The formula for distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Substituting the values:

• $(x_1, y_1) = (-6, 4)$
• $(x_2, y_2) = (-3, 6)$

We get:

$r = \sqrt{((-3) - (-6))^2 + (6 - 4)^2}$ $r = \sqrt{(3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13}$

Thus, the radius is $\sqrt{13}$.

Now, we can write the equation of the circle as:

$(x + 6)^2 + (y - 4)^2 = 13$

This is the equation of the circle.

Would you like more details or have any questions on this? Here are some related questions you might find helpful:

1. How do you calculate the distance between two points on a coordinate plane?
2. What is the standard form of a circle's equation?
3. How do you graph a circle from its equation?
4. What are other ways to find the equation of a circle besides using the center and radius?
5. How can you determine the center and radius of a circle from its equation?

Tip: Always check the signs of the center’s coordinates when writing the equation of the circle; they should match the opposite of what's in the formula!