Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. (x + 1)² + y² = 25

Verified step by step guidance

1. Recognize that the given equation is in the standard form of a circle: \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.
2. Rewrite the equation \( (x + 1)^2 + y^2 = 25 \) as \( (x - (-1))^2 + (y - 0)^2 = 5^2 \) to identify the center and radius clearly.
3. From the rewritten form, identify the center of the circle as \( (-1, 0) \) and the radius as \( 5 \).
4. To find the domain of the circle, consider the horizontal distance from the center: the domain is all \(x\) values such that \(x\) is between \(-1 - 5\) and \(-1 + 5\), or \([-6, 4]\).
5. To find the range of the circle, consider the vertical distance from the center: the range is all \(y\) values such that \(y\) is between \$0 - 5\( and \)0 + 5\(, or \)[-5, 5]$.

Key Concepts

Standard Form of a Circle's Equation

The standard 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. Recognizing this form allows you to identify the circle's center and radius directly from the equation.

Graphing Circles

Graphing a circle involves plotting its center and using the radius to mark points equidistant from the center in all directions. This visual representation helps in understanding the shape and position of the circle on the coordinate plane.

Domain and Range of a Circle

The domain of a circle is the set of all possible x-values, and the range is the set of all possible y-values covered by the circle. For a circle centered at (h, k) with radius r, the domain is [h - r, h + r] and the range is [k - r, k + r].