%5E2%2By%5E2%3D1-graph.jpeg "(x-1)^2+y^2=1 Graph")
Understanding the Graph of (x-1)² + y² = 1
The equation (x-1)² + y² = 1 represents a circle in the Cartesian coordinate system. Let's break down how to visualize this graph:
Key Concepts
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Standard Circle Equation: The general form of a circle equation is (x - h)² + (y - k)² = r², where:
- (h, k) represents the center of the circle.
- r represents the radius of the circle.
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Our Equation: In our case, (x - 1)² + y² = 1, we can see:
- The center of the circle is (1, 0) because h = 1 and k = 0.
- The radius of the circle is 1 because r² = 1, so r = 1.
Graphing the Circle
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Locate the Center: Plot the point (1, 0) on the coordinate plane.
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Draw the Radius: From the center (1, 0), move 1 unit to the right, 1 unit to the left, 1 unit up, and 1 unit down. These points represent the intersections of the circle with the x and y axes.
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Connect the Points: Connect these four points with a smooth curve to form the circle.
Key Features of the Graph
- Symmetry: The circle is symmetric about both the x and y axes.
- Radius: The distance from the center to any point on the circle is always 1 unit.
- Circumference: The distance around the circle is given by 2πr, which in this case is 2π.
- Area: The area enclosed by the circle is given by πr², which in this case is π.
Example:
Imagine a circle on a graph paper with its center at the point (1,0). Now, draw a line segment of length 1 unit starting from the center (1,0) in any direction. You will be able to draw another point on the circle. Doing this repeatedly and connecting all the points will result in a circle.
Conclusion
Understanding the standard circle equation and the key features of the graph helps us visualize and analyze the equation (x-1)² + y² = 1. By plotting the center and radius, we can accurately depict the circle on the coordinate plane. This equation provides a foundational understanding for further exploration of circles and their properties in geometry and other mathematical fields.