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

Verified step by step guidance

1. Step 1: Recognize that the equation given, \((x - 3)^2 + (y - 2)^2 = 25\), represents a circle in the coordinate plane. This is because it is in the standard form of a circle's equation: \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
2. Step 2: Identify the center of the circle from the equation. Here, the center is at the point \((3, 2)\) because the equation is \((x - 3)^2 + (y - 2)^2 = 25\).
3. Step 3: Determine the radius of the circle by taking the square root of the right side of the equation. Since \(25\) is \(r^2\), the radius \(r\) is \(\sqrt{25}\).
4. Step 4: Use the center and radius information to match the equation to the correct graph in Column II. Look for the graph that shows a circle centered at \((3, 2)\) with radius equal to \(r\).
5. Step 5: Repeat this process for the other equations in Column I by identifying their forms (circle, ellipse, parabola, hyperbola, etc.) and matching their key features (center, vertices, axes lengths) to the graphs in Column II.

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 from visual data. Being able to analyze the graph's features allows you to connect it accurately to its corresponding equation.

Coordinate Geometry Basics

Coordinate geometry principles, such as plotting points and understanding distances between points, are fundamental for visualizing and verifying the relationship between algebraic equations and their graphical representations.