Graph the function y = √|x|.

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Start by understanding the function y = √|x|. This function involves the square root and absolute value, which affects its graph. The absolute value ensures that the input x is non-negative, and the square root function is defined for non-negative values.

Consider the domain of the function. Since the square root function is only defined for non-negative numbers, and the absolute value ensures x is non-negative, the domain of y = √|x| is all real numbers, x ∈ ℝ.

Analyze the behavior of the function for different values of x. For x ≥ 0, the function simplifies to y = √x. For x < 0, the function becomes y = √(-x), which is equivalent to y = √|x| due to the absolute value.

Plot key points to understand the shape of the graph. For example, when x = 0, y = √|0| = 0. When x = 1, y = √|1| = 1. When x = -1, y = √|-1| = 1. Notice that the graph is symmetric about the y-axis.

Sketch the graph using the points and symmetry observed. The graph will have a V-shape, opening upwards, with the vertex at the origin (0,0). The left and right branches will be identical, reflecting the symmetry due to the absolute value.

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Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. For any real number x, |x| is x if x is positive or zero, and -x if x is negative. This concept is crucial for understanding how the function y = √|x| behaves symmetrically around the y-axis.

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Square Root Function

The square root function, denoted as √x, returns the non-negative value whose square is x. It is defined for x ≥ 0 and results in a curve that increases slowly as x increases. In the context of y = √|x|, the square root is applied to the absolute value, ensuring the function is defined for all real x.

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Graphing Functions

Graphing functions involves plotting points that satisfy the function's equation on a coordinate plane. For y = √|x|, understanding the behavior of both the absolute value and square root is essential. The graph will be a V-shaped curve, symmetric about the y-axis, starting at the origin and opening upwards.

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