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(x^2+y^2)^(1/2) Graph

3 min read Jun 17, 2024
(x^2+y^2)^(1/2) Graph

Understanding the Graph of (x^2 + y^2)^(1/2)

The expression (x^2 + y^2)^(1/2) represents the distance formula in two dimensions. It calculates the distance between a point (x, y) and the origin (0, 0). Let's break down why this is the case and how it translates to a graph.

The Distance Formula

Remember the Pythagorean theorem? It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In our case:

  • Hypotenuse: The distance between the origin (0, 0) and the point (x, y). This is what our expression (x^2 + y^2)^(1/2) calculates.
  • Other sides: The horizontal distance (x) and the vertical distance (y) from the origin to the point (x, y).

Therefore, the equation (x^2 + y^2)^(1/2) represents the length of the hypotenuse, which is the distance between the point (x, y) and the origin (0, 0).

The Graph

The graph of (x^2 + y^2)^(1/2) is a circle centered at the origin with a radius of 1. Here's why:

  1. Distance is constant: For any point (x, y) on the graph, the expression (x^2 + y^2)^(1/2) will always equal 1. This means that the distance from the origin to any point on the graph is always 1.

  2. Circle definition: A circle is defined as the set of all points that are equidistant from a fixed point (the center). In our case, the fixed point is the origin, and the constant distance is 1.

Summary

In essence, the graph of (x^2 + y^2)^(1/2) represents the set of all points in the plane that are 1 unit away from the origin. This defines a circle with a radius of 1, centered at the origin.

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