Concept Check Match each equation in Column I with its graph in Column II. I II 47. A. 48. B. 49. C. 50. (x + 3)² + (y + 2)² = 25 D.

Verified step by step guidance

1. Step 1: Identify the type of equation given in Column I. The equation \( (x + 3)^2 + (y + 2)^2 = 25 \) represents a circle because it is in the standard form of a circle equation \( (x - h)^2 + (y - k)^2 = r^2 \).
2. Step 2: From the equation \( (x + 3)^2 + (y + 2)^2 = 25 \), determine the center and radius of the circle. The center is at \( (-3, -2) \) and the radius is \( \sqrt{25} = 5 \).
3. Step 3: Examine the graphs in Column II and look for the graph that shows a circle centered at \( (-3, -2) \) with radius 5. This graph will match the equation given.
4. Step 4: For the other equations in Column I (A, B, C), identify their types (e.g., lines, parabolas, ellipses, etc.) by analyzing their algebraic form and compare with the graphs in Column II to find the correct matches.
5. Step 5: Match each equation with its corresponding graph by verifying key features such as intercepts, vertex, center, radius, or slope, depending on the type of equation.

Key Concepts

Equation of a Circle

The equation (x + h)² + (y + k)² = r² represents a circle centered at (-h, -k) with radius r. Understanding this form helps identify the circle's position and size on the coordinate plane, which is essential for matching equations to their graphs.

Graph Interpretation

Interpreting graphs involves recognizing shapes, centers, and radii of circles or other curves. Being able to visually analyze graphs and relate them to their algebraic equations is crucial for correctly matching each equation to its corresponding graph.

Coordinate Geometry Basics

Coordinate geometry connects algebraic equations with geometric figures on the Cartesian plane. Familiarity with plotting points, understanding shifts in the center, and measuring distances aids in linking equations like circles to their graphical representations.