Exercises 103–105 will help you prepare for the material covered in the next section. Use a rectangular coordinate system to graph the circle with center (1, -1) and radius 1.

검증된 단계별 안내

Recall the standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]

Identify the center \((h, k)\) and radius \(r\) from the problem: here, \(h = 1\), \(k = -1\), and \(r = 1\).

Substitute these values into the standard form equation to write the specific equation of the circle: \[ (x - 1)^2 + (y + 1)^2 = 1^2 \]

To graph the circle, first plot the center point at \((1, -1)\) on the rectangular coordinate system.

From the center, use the radius to mark points 1 unit away in all four directions (up, down, left, right), then sketch the circle passing through these points.

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주요 개념

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

Equation of a Circle

The equation of a circle with center (h, k) and radius r is given by (x - h)² + (y - k)² = r². This formula represents all points (x, y) that are exactly r units away from the center. Understanding this equation is essential for graphing the circle accurately.

Coordinate Plane and Plotting Points

A rectangular coordinate system consists of an x-axis and y-axis intersecting at the origin. Plotting points involves locating coordinates (x, y) on this plane. Knowing how to plot the center and points on the circle helps visualize and draw the circle correctly.

Radius and Distance in the Plane

The radius is the fixed distance from the center of the circle to any point on its circumference. Understanding how to measure and use this distance in the coordinate plane allows you to determine points on the circle by moving r units horizontally or vertically from the center.