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Chapter 1: Problem 78

Graph the function \(y=\sqrt{|x|}\)

Short Answer

Expert verified
The graph of \(y = \sqrt{|x|}\) is a V-shaped curve symmetric around the y-axis.

Step by step solution

01

Analyze the Function

The function is given as \(y = \sqrt{|x|}\). Notice that the function involves an absolute value and a square root. The expression inside the square root, \(|x|\), means the function is defined for all real numbers \(x\), because the square root of a non-negative number is always valid.
02

Domain and Range

Determine the domain and range of the function. Since the square root can only handle non-negative values and \(|x|\) is always non-negative, the domain of the function is all real numbers \(x \in \mathbb{R}\). The range, being the result of the square root, is all non-negative real numbers: \(y \geq 0\).
03

Identify Key Points

Identify key points that will help in sketching the graph. When \(x=0\), \(y=\sqrt{|0|} = 0\). When \(x=1\), \(y=\sqrt{|1|} = 1\), and similarly, when \(x=-1\), \(y=\sqrt{|-1|}=1\). Consider another point, like \(x=4\), for both positive and negative values, e.g., \(x=4\) and \(x=-4\), both giving \(y = 2\).
04

Symmetry

Understand that the graph is symmetric around the \(y\)-axis because the absolute value \(|x|\) causes any negative \(x\) to behave like its positive counterpart. This means the graph for positive \(x\) can be mirrored along the \(y\)-axis for negative \(x\).
05

Sketch the Graph

Use the identified points and properties to sketch the graph. Place the key points \((0,0), (1,1), (-1,1), (4,2), (-4,2)\) and draw a curve smoothly connecting them. The graph approaches the \(x\)-axis as \(x\) moves far from zero, forming a V-shaped curve. It will look like the right half of a parabola reflecting over the \(y\)-axis.
06

Verify the Graph

Double-check that all characteristics have been considered: the domain \(x \in \mathbb{R}\), range \(y \geq 0\), symmetry about the \(y\)-axis, and the shape of the graph. Ensure the points \((0,0), (1,1), (-1,1), (4,2), (-4,2)\) fit well into the graph constructed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain and Range
When discussing the domain and range of a function, it's crucial to understand where the function is defined and what values it can take. For the function \(y = \sqrt{|x|}\), let's begin by analyzing the domain:
  • The absolute value, \(|x|\), means the expression inside the square root is never negative.
  • Since square roots of non-negative numbers are defined, any real number \(x\) is allowed.
Therefore, the domain of \(y = \sqrt{|x|}\) is all real numbers \(x \in \mathbb{R}\).

Next, let's consider the range:
  • The output, \(y\), is the result of a square root, which is always non-negative.
  • This means \(y\) can take any value from 0 and upwards.
Thus, the range of the function is all non-negative real numbers: \(y \geq 0\).

Having a clear sense of domain and range can help you graph the function effectively.
Symmetry
Symmetry in graphs helps to understand the overall shape by using patterns. With the function \(y = \sqrt{|x|}\), symmetry can be explored in relation to the \(y\)-axis:
  • Since the function includes an absolute value, \(|x|\), both positive and negative \(x\) yield the same results, making the graph reflect across the \(y\)-axis.
  • This means for every point \((x, y)\) on the graph, there will be a corresponding point \((-x, y)\).
Understanding this symmetry allows you to mirror the shape on one side of the \(y\)-axis to the other, creating a balanced graph.

With this knowledge, it's easy to predict and verify the behavior of the graph on either side of the \(y\)-axis.
Absolute Value
The concept of absolute value, \(|x|\), is integral to the function \(y = \sqrt{|x|}\). Absolute value measures the distance of a number from zero on the number line, regardless of direction.

Here are some essential points:
  • \(|x|\) makes every input non-negative.
  • This property ensures that \(\sqrt{\cdot}\) is always applied to a non-negative number, leading to a valid square root for any real number \(x\).
For instance:
  • When \(x = -2\), \(|x| = 2\).
  • When \(x = 3\), \(|x| = 3\).
These transformations ensure that the function behaves similarly for both negative and positive inputs, paving the way for the symmetric nature of the graph, as discussed earlier.
Square Root Function
The square root function \(y = \sqrt{|x|}\) has specific characteristics that influence the overall graph. Understanding these properties helps in constructing and analyzing the shape:
  • The square root operation is defined for all non-negative inputs, reflected by the range \(y \geq 0\).
  • The function gradually increases from zero because as \(|x|\) grows, \(y = \sqrt{|x|}\) increases.
  • It generates a curve that is slower to rise compared to linear functions.
For example,\(\sqrt{|x|}\) moves upward quickly at first but then rises more slowly as \(|x|\) becomes larger.

This slow rise is evident in the key points such as:
  • \((0, 0)\) where the increase begins.
  • \((1, 1)\) and \((4, 2)\) illustrating the gentle slope of the graph.
The overall shape resembles a part of a parabola, but smoother, creating a gentle, upward-opening curve when plotted, characterizing the nature of square root functions.

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