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 - 2)² = 4

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Identify the given equation of the circle: \( (x+3)^2 + (y+2)^2 = 4 \). This is in the standard 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.

Compare the given equation to the standard form to find the center. Since the equation is \( (x + 3)^2 + (y + 2)^2 = 4 \), rewrite it as \( (x - (-3))^2 + (y - (-2))^2 = 2^2 \). Therefore, the center is at \( (-3, -2) \).

Determine the radius by taking the square root of the right side of the equation. Here, \( r^2 = 4 \), so \( r = 2 \).

To graph the circle, plot the center at \( (-3, -2) \) on the coordinate plane. Then, draw a circle with radius 2 units around this center, marking points 2 units away in all directions (up, down, left, right).

Identify the domain and range from the graph or equation. The domain is all \( x \)-values covered by the circle, which is from \( h - r \) to \( h + r \), so from \( -3 - 2 \) to \( -3 + 2 \). The range is all \( y \)-values covered by the circle, from \( k - r \) to \( k + r \), so from \( -2 - 2 \) to \( -2 + 2 \).

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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. By comparing the given equation to this form, you can identify the circle's center and radius directly.

Graphing Circles

Graphing a circle involves plotting its center and using the radius to mark points 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 x-values covered by the circle, and the range is the set of all y-values. These can be found by considering the center coordinates and radius, as the circle extends r units horizontally and vertically from the center.