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

Verified step by step guidance

1. Identify the type of each equation in Column I. For example, recognize that an equation like \((x + 3)^2 + (y - 2)^2 = 25\) represents a circle with center at \((-3, 2)\) and radius \(5\) (since \(25\) is \$5^2$).
2. Recall the general forms of common graphs: circles, parabolas, ellipses, and hyperbolas, and their key features such as center, vertices, axes, and intercepts.
3. Analyze each graph in Column II by noting its shape and key points, such as the center of a circle or the vertex of a parabola, and compare these features to the equations in Column I.
4. Match each equation to the graph that corresponds to its shape and key characteristics. For example, the circle equation should match the graph showing a circle centered at \((-3, 2)\) with radius \(5\).
5. Double-check your matches by verifying that the graph's features (like intercepts or symmetry) align with the properties derived from the equation.

Key Concepts

Equation of a Circle

The equation (x + 3)² + (y - 2)² = 25 represents a circle centered at (-3, 2) with a radius of 5. Understanding this standard form helps identify the graph as a circle and locate its center and size.

Graph Matching Techniques

Matching equations to graphs requires recognizing key features such as shape, intercepts, and transformations. This skill involves analyzing the equation's form and comparing it to visual characteristics of the graphs.

Coordinate Geometry and Transformations

Understanding how shifts in x and y (like +3 or -2 inside the equation) translate the graph horizontally or vertically is essential. This concept helps in predicting the graph's position relative to the origin.