To write the standard form of the equation of the circle shown in the image, we use the general formula for the equation of a circle:

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

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

Step 1: Identify the center of the circle

From the graph, we can see that the center of the circle is at $(-3, 3)$.

Step 2: Calculate the radius

The radius is the distance from the center to any point on the circle. From the graph, the point $(5, 3)$ lies on the circle. The radius is the horizontal distance between the center $(-3, 3)$ and this point. We calculate the distance between $(-3, 3)$ and $(5, 3)$:

$r = 5 - (-3) = 5 + 3 = 8$

Thus, the radius $r = 8$.

Step 3: Write the equation

Now we substitute the values of $h = -3$, $k = 3$, and $r = 8$ into the standard form equation:

$(x + 3)^2 + (y - 3)^2 = 8^2$

Simplifying:

$(x + 3)^2 + (y - 3)^2 = 64$

So, the standard form of the equation of the circle is:

$(x + 3)^2 + (y - 3)^2 = 64$

Let me know if you need further details or explanations.

Here are 5 related questions you can explore:

1. How do you find the equation of a circle given its diameter endpoints?
2. How do you derive the equation of a circle from its general form to standard form?
3. How can you calculate the radius when given the area of a circle?
4. What is the distance formula and how is it used in circle problems?
5. How do you determine if a point lies inside, outside, or on a circle based on the equation?

Tip: Always check the coordinates of the center and the radius carefully from the graph to avoid errors when writing the equation.