Graph each inequality. x² + (y + 3)² ≤ 16

Verified step by step guidance

1. Identify the inequality as representing a circle with all points inside or on the boundary. The general 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.
2. Rewrite the given inequality \( x^2 + (y + 3)^2 \leq 16 \) in the form \( (x - 0)^2 + (y - (-3))^2 \leq 4^2 \). This shows the center is at \((0, -3)\) and the radius is \(4\).
3. Draw the circle centered at \((0, -3)\) with radius \(4\) on the coordinate plane. This circle includes all points \((x, y)\) that satisfy the equation \( (x - 0)^2 + (y + 3)^2 = 16 \).
4. Since the inequality is \( \leq \), shade the entire region inside and on the circle to represent all points where \( x^2 + (y + 3)^2 \leq 16 \) holds true.
5. Label the center and radius on the graph for clarity, and ensure the boundary circle is solid (not dashed) because points on the circle satisfy the inequality.

Key Concepts

Graphing Circles

The inequality x² + (y + 3)² ≤ 16 represents all points inside or on the circle centered at (0, -3) with radius 4. Understanding how to graph circles involves identifying the center and radius from the equation and plotting the boundary circle accordingly.

Inequalities and Regions

Inequalities like ≤ indicate that the solution includes all points inside the boundary curve, not just on it. For this problem, shading the region inside the circle shows all points (x, y) satisfying the inequality.

Coordinate Geometry and Transformations

Recognizing the shift in the circle’s center from the origin to (0, -3) involves understanding how adding or subtracting values inside the equation translates the graph vertically or horizontally in the coordinate plane.