In the image, we are asked to complete the equation of the circle. Based on the diagram, we can extract some important information to do this.

1. Center of the Circle: The point $A$, marked in the graph, is the center of the circle. From the grid, we can see that the coordinates of point $A$ are $(1, 1)$.

2. Radius of the Circle: By inspecting the distance from the center $A(1, 1)$ to the edge of the circle, we can count one unit horizontally or vertically to the edge of the circle, meaning the radius $r$ of the circle is $1$.

Now, the standard equation of a circle with center $(h, k)$ and radius $r$ is:

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

Substituting the values for $h = 1$, $k = 1$, and $r = 1$, we get the equation:

$(x - 1)^2 + (y - 1)^2 = 1^2$

Which simplifies to:

$(x - 1)^2 + (y - 1)^2 = 1$

So, the completed equation of the circle is:

$(x - 1)^2 + (y - 1)^2 = 1$

Would you like further details on any step, or do you have any questions?

Here are 5 related questions you might find helpful:

1. How can we find the center of a circle from its equation?
2. How is the radius of a circle related to its equation?
3. Can the equation of a circle have a different form for a circle not centered at the origin?
4. How do we derive the general form of the circle equation from the standard form?
5. What happens if the radius is not a whole number?

Tip:

Always make sure to square the radius correctly in the final equation, as small errors can impact the solution!