Use each graph to determine an equation of the circle in (a) center-radius form and (b) general form.

Verified step by step guidance

1. Identify the center \((h, k)\) of the circle from the graph. This is the point 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.
3. Write the equation of the circle in center-radius form using the formula: \[ (x - h)^2 + (y - k)^2 = r^2 \]
4. Expand the squared terms in the center-radius form to convert it into the general form. This involves expanding \((x - h)^2\) and \((y - k)^2\).
5. Simplify the expanded expression and rearrange all terms to one side to write the equation in general form: \[ x^2 + y^2 + Dx + Ey + F = 0 \] where \(D\), \(E\), and \(F\) are constants derived from the expansion.

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 shows the circle's center coordinates and radius, making it easy to write the equation when these values are known from the graph.

Converting to General Form of a Circle

The general form of a circle's equation is x² + y² + Dx + Ey + F = 0. It is obtained by expanding the center-radius form and simplifying. Understanding how to expand and rearrange terms is essential to rewrite the equation in this standard polynomial form.

Interpreting Graphs to Identify Circle Parameters

Analyzing a graph of a circle involves identifying the center point and measuring the radius, which is the distance from the center to any point on the circle. Accurate reading of these values from the graph is crucial for writing the correct equation in both forms.