Concept Check Match each equation in Column I with its graph in Column II. I II47. A. 48. B. 49. C. 50. (x + 3)² + (y + 2)² = 25 D.

Verified step by step guidance

1. Identify the type of equation given in Column I. The equation \((x + 3)^2 + (y + 2)^2 = 25\) is a circle equation in standard form \((x - h)^2 + (y - k)^2 = r^2\).
2. Determine the center of the circle from the equation. The center \((h, k)\) is \((-3, -2)\) because the equation is \((x + 3)^2 + (y + 2)^2 = 25\).
3. Find the radius of the circle. The radius \(r\) is the square root of 25, which is 5.
4. Match the equation with its graph in Column II. Look for a graph that represents a circle centered at \((-3, -2)\) with a radius of 5.
5. Verify the match by checking if the graph in Column II corresponds to the circle's center and radius identified from the equation.

Key Concepts

Circle Equation

The equation (x + 3)² + (y + 2)² = 25 represents a circle in the Cartesian plane. The center of the circle is at the point (-3, -2), and the radius is the square root of 25, which is 5. Understanding this standard form of a circle's equation is crucial for identifying its graph.

Graphing Techniques

Graphing techniques involve plotting points and understanding the shape of various equations. For a circle, one must recognize that all points on the graph are equidistant from the center. This concept is essential for accurately matching the equation to its corresponding graph.

Coordinate System

The coordinate system is a two-dimensional plane defined by the x-axis and y-axis. Each point on this plane is represented by an ordered pair (x, y). Familiarity with this system is necessary for interpreting the position of the circle's center and its radius in relation to the axes.