Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 1 - Functions

Problem 1.2.78

Chapter 1, Problem 1.2.78

Graph the function y = √|x|.

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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sin² θ = cos² θ

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Textbook Question

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identify A, B, C, and D for the sine functions in Exercises 67–70 and sketch their graphs.

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