Exploring the Semi-Circle Function | Properties, Equation, and Applications

Semi-Circle Function

The semi-circle function, also known as the half-circle function, is a mathematical function that represents a semi-circle graph

The semi-circle function, also known as the half-circle function, is a mathematical function that represents a semi-circle graph. It is defined as a curve that has the shape of half of a circle.

The equation of the semi-circle function can be written in different forms depending on the specific properties desired. One common form is:
y = √(r^2 – x^2)

In this equation:
– y represents the height or value of the function at a given x-coordinate.
– x represents the x-coordinate along the horizontal axis.
– r represents the radius of the semi-circle.

The semi-circle function is commonly used in geometry and trigonometry to model and solve various problems. It also has applications in physics, engineering, and computer graphics.

Properties of the semi-circle function include:
1. Domain and Range: The domain of the semi-circle function is typically restricted to values between -r and r, as the x-coordinate cannot exceed the radius of the semi-circle. The range is typically between 0 and r, as the y-coordinate represents the height of the semi-circle.
2. Symmetry: The semi-circle function is symmetric about the y-axis. This means that if a point (x, y) lies on the semi-circle, then the point (-x, y) also lies on the semi-circle.
3. Center and Diameter: The center of the semi-circle is located at the origin, (0, 0). The diameter of the semi-circle is twice the radius, and it is a horizontal line passing through the center.
4. Intercepts: The semi-circle intersects the x-axis at two points: (-r, 0) and (r, 0). These points are also called the x-intercepts. The semi-circle does not intersect the y-axis, as its center lies on the y-axis.
5. Area: The area enclosed by the semi-circle can be calculated using the formula: A = πr^2/2, where A is the area and r is the radius.

In summary, the semi-circle function is a mathematical representation of a half-circle graph. It has specific properties, such as symmetry, intercepts, and a defined area, which make it useful in various mathematical and real-world applications.

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