To find the center and radius of the circle given by the equation:

$(x + 4)^2 + (y - 1)^2 = 4$

we need to identify it in the standard form of a circle's equation:

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

Step-by-Step Solution

1. Identify the Center $(h, k)$:

   • In the equation $(x + 4)^2 + (y - 1)^2 = 4$, we have $(x - (-4))^2$ and $(y - 1)^2$.
   • This tells us the center $(h, k)$ is at $(-4, 1)$.

2. Find the Radius $r$:

   • The right side of the equation is 4, which is $r^2$.
   • Solving for $r$, we get $r = \sqrt{4} = 2$.

Answer

• Center: $(-4, 1)$
• Radius: $2$

Now, to graph the circle:

1. Plot the center at $(-4, 1)$.
2. Draw a circle with a radius of 2 units from the center, ensuring it spans 2 units in all directions around the center.

Would you like further explanation on graphing or any other details?

Follow-up Questions

1. How can you rewrite an equation to identify the center and radius of a circle?
2. What does changing the values of $h$ and $k$ do to the position of the circle?
3. How is the radius calculated when given an equation of a circle?
4. What is the significance of the radius in the graph of a circle?
5. How would you graph a circle with a center at the origin?

Tip:

When identifying the radius in a circle equation, remember that $r$ is the square root of the constant on the right side.