Give the center and radius of each circle and graph.
$(x+2{)}^2+(y-3{)}^2=4$

• A. $r=2$; Center: $(3,-2)$
• B. $r=2$; Center: $(3,-2)$
• C. $r=2$; Center: $\left(-2,3\right)$
• D. $r=2$; Center: $(-2,3)$
  

Answer: (D)

Verified step by step guidance

1. Identify the general form of the equation of a circle: \(\left(x - h\right)^2 + \left(y - k\right)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
2. Look at the given equation: \(\left(x + 2\right)^2 + \left(y - 3\right)^2 = 4\). Rewrite it to match the general form: \(\left(x - (-2)\right)^2 + \left(y - 3\right)^2 = 2^2\).
3. From the rewritten equation, identify the center as \((-2, 3)\) because the signs inside the parentheses are opposite to the signs in the center coordinates.
4. Identify the radius \(r\) by taking the square root of the right side of the equation: \(r = \sqrt{4} = 2\).
5. To graph the circle, plot the center at \((-2, 3)\) on the coordinate plane, then use the radius to mark points 2 units away in all directions (up, down, left, right) from the center, and draw a smooth curve connecting these points to form the circle.