Connecting Graphs with Equations Use each graph to determine an equation of the circle in center-radius form.

Verified step by step guidance

1. Identify the center of the circle from the graph. The center is the point \((h, k)\) where the circle is centered.
2. Determine the radius \(r\) of the circle by measuring the distance from the center to any point on the circle's edge.
3. Recall the center-radius 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.
4. Substitute the values of \(h\), \(k\), and \(r\) into the equation to write the specific equation of the circle.
5. Double-check your equation by verifying that points on the circle satisfy the equation when substituted for \(x\) and \(y\).

Key Concepts

Equation of a Circle in Center-Radius Form

The center-radius form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form directly relates the geometric properties of the circle to its algebraic representation, making it easier to write the equation from a graph.

Identifying the Center from a Graph

The center of a circle on a graph is the point equidistant from all points on the circle's circumference. It can be found by locating the midpoint of the diameter or by observing the symmetry of the circle along the x and y axes.

Determining the Radius from a Graph

The radius is the distance from the center of the circle to any point on its edge. On a graph, this can be measured by counting the units between the center and a point on the circle along the x or y axis, or by using the distance formula if coordinates are known.