In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. center (-2, 5), radius 4

Verified step by step guidance

1. Recall that the center-radius form of a circle's equation is given by the formula: \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius.
2. Identify the center \( (h, k) \) and the radius \( r \) from the problem. Here, the center is \( (-2, 5) \) and the radius is \( 4 \).
3. Substitute the center coordinates and the radius into the center-radius form: replace \( h \) with \( -2 \), \( k \) with \( 5 \), and \( r \) with \( 4 \).
4. Write the equation as \( (x - (-2))^2 + (y - 5)^2 = 4^2 \), which simplifies to \( (x + 2)^2 + (y - 5)^2 = 16 \).
5. For graphing, plot the center at \( (-2, 5) \) on the coordinate plane, then draw a circle with radius 4 units around this point, marking points 4 units away in all directions (up, down, left, right).

Key Concepts

Equation of a Circle in Center-Radius Form

The center-radius form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. This form directly shows the circle's location and size, making it easier to graph and analyze.

Identifying the Center and Radius

Given the center coordinates and radius, you substitute these values into the center-radius formula. For example, with center (-2, 5) and radius 4, the equation becomes (x + 2)^2 + (y - 5)^2 = 16, since r^2 = 4^2 = 16.

Graphing a Circle

To graph a circle, plot the center point first, then use the radius to mark points in all directions (up, down, left, right) from the center. Connect these points smoothly to form the circle, ensuring the radius distance is consistent.