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 − 3)² + (y + 1)² = 36

검증된 단계별 안내

Recognize that the given equation is in the standard form of a circle: \(\left(x - h\right)^2 + \left(y - k\right)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Identify the center \((h, k)\) by comparing the given equation \(\left(x - 3\right)^2 + \left(y + 1\right)^2 = 36\) to the standard form. Note that \(y + 1\) can be rewritten as \(y - (-1)\), so the center is at \((3, -1)\).

Determine the radius \(r\) by taking the square root of the right side of the equation: \(r = \sqrt{36}\).

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 \(h - r\) and \(h + r\), or \(3 - r \leq x \leq 3 + r\).

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 \(k - r\) and \(k + r\), or \(-1 - r \leq y \leq -1 + r\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Equation of a Circle

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. This form helps identify the circle's position and size directly from the equation.

Domain and Range of a Circle

The domain and range of a circle are the sets of possible x and y values, respectively. 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], representing all points covered by the circle horizontally and vertically.

Graphing Circles

Graphing a circle involves plotting its center and using the radius to mark points in all directions. This visual representation helps confirm the circle's shape and aids in identifying the domain and range by observing the extent of the graph along the x- and y-axes.