In Exercises 41–52, 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. x² + (y − 1)² = 1

Verified step by step guidance

1. Step 1: Recognize the standard form of a circle's equation, which is (x − h)² + (y − k)² = r², where (h, k) is the center of the circle and r is the radius.
2. Step 2: Compare the given equation x² + (y − 1)² = 1 to the standard form. Notice that h = 0 (since x² is the same as (x − 0)²), k = 1 (from (y − 1)²), and r² = 1.
3. Step 3: Determine the center of the circle. From the comparison, the center is (h, k) = (0, 1).
4. Step 4: Find the radius of the circle. Since r² = 1, take the square root of both sides to find r = √1 = 1.
5. Step 5: To graph the circle, plot the center at (0, 1) and draw a circle with a radius of 1. The domain of the circle is the set of x-values from -1 to 1, and the range is the set of y-values from 0 to 2.

Key Concepts

Circle 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. In the given equation, x² + (y - 1)² = 1, we can identify the center as (0, 1) and the radius as 1, since r² = 1.

Graphing Circles

To graph a circle, plot the center point and use the radius to mark points in all directions (up, down, left, right) from the center. The resulting shape is a circle, and understanding how to accurately represent this visually is crucial for analyzing the relation's domain and range.

Domain and Range

The domain of a relation refers to all possible x-values, while the range refers to all possible y-values. For the circle described, the domain is [-1, 1] and the range is [0, 2], as these intervals encompass all x and y coordinates that the circle can reach based on its center and radius.