Understanding the Absolute Value Part

Oct 13, 2025 by windailys.com 118 views

Introduction

Absolute value functions are a fundamental concept in mathematics, and they play a crucial role in various mathematical operations. In this article, we will focus on expressing the function $y = |x - 4| - 2$ and explore its properties and behavior.

What is an Absolute Value Function?

An absolute value function is a mathematical function that returns the absolute value of a given expression. The absolute value of a number is its distance from zero on the number line, without considering its direction. In other words, it is the non-negative value of a number.

The Function $y = |x - 4| - 2$

The given function is $y = |x - 4| - 2$ . To understand this function, let's break it down into two parts:

The absolute value part: $|x - 4|$
The constant part: $-2$

Understanding the Absolute Value Part

The absolute value part, $|x - 4|$ , represents the distance between $x$ and $4$ on the number line. When $x$ is greater than $4$ , the distance is $x - 4$ , and when $x$ is less than $4$ , the distance is $4 - x$ .

Understanding the Constant Part

The constant part, $-2$ , represents a vertical shift of the graph of the absolute value function. This means that the graph of the function will be shifted down by $2$ units.

Expressing the Function

To express the function $y = |x - 4| - 2$ , we need to consider two cases:

Case 1: $x \geq 4$
Case 2: $x < 4$

Case 1: $x \geq 4$

When $x \geq 4$ , the absolute value part, $|x - 4|$ , simplifies to $x - 4$ . Therefore, the function becomes:

y = (x - 4) - 2

Simplifying this expression, we get:

y = x - 6

Case 2: $x < 4$

When $x < 4$ , the absolute value part, $|x - 4|$ , simplifies to $4 - x$ . Therefore, the function becomes:

y = (4 - x) - 2

Simplifying this expression, we get:

y = 2 - x

Graphing the Function

The graph of the function $y = |x - 4| - 2$ consists of two parts:

A line segment with a slope of $1$ and a y-intercept of $-6$ for $x \geq 4$
A line segment with a slope of $-1$ and a y-intercept of $2$ for $x < 4$

Properties of the Function

The function $y = |x - 4| - 2$ has the following properties:

Domain: The domain of the function is all real numbers, $(-\infty, \infty)$ .
Range: The range of the function is all real numbers, $(-\infty, \infty)$ .
Symmetry: The function is symmetric about the line $x = 4$ .
Maximum and Minimum: The function has a maximum value of $-2$ at $x = 4$ and a minimum value of $-6$ at $x = \infty$ .

Conclusion

Q: What is the absolute value function?

A: The absolute value function is a mathematical function that returns the absolute value of a given expression. The absolute value of a number is its distance from zero on the number line, without considering its direction.

Q: What is the difference between the absolute value function and the absolute value expression?

A: The absolute value function is a mathematical function that takes an expression as input and returns its absolute value. The absolute value expression, on the other hand, is a mathematical expression that involves the absolute value of a variable or a constant.

Q: How do I express the function $y = |x - 4| - 2$ ?

A: To express the function $y = |x - 4| - 2$ , you need to consider two cases:

Case 1: $x \geq 4$
Case 2: $x < 4$

For Case 1, the function becomes $y = x - 6$ . For Case 2, the function becomes $y = 2 - x$ .

Q: What is the domain of the function $y = |x - 4| - 2$ ?

A: The domain of the function $y = |x - 4| - 2$ is all real numbers, $(-\infty, \infty)$ .

Q: What is the range of the function $y = |x - 4| - 2$ ?

A: The range of the function $y = |x - 4| - 2$ is all real numbers, $(-\infty, \infty)$ .

Q: Is the function $y = |x - 4| - 2$ symmetric about the line $x = 4$ ?

A: Yes, the function $y = |x - 4| - 2$ is symmetric about the line $x = 4$ .

Q: What is the maximum value of the function $y = |x - 4| - 2$ ?

A: The maximum value of the function $y = |x - 4| - 2$ is $-2$ , which occurs at $x = 4$ .

Q: What is the minimum value of the function $y = |x - 4| - 2$ ?

A: The minimum value of the function $y = |x - 4| - 2$ is $-6$ , which occurs at $x = \infty$ .

Q: How do I graph the function $y = |x - 4| - 2$ ?

A: To graph the function $y = |x - 4| - 2$ , you need to plot two line segments:

A line segment with a slope of $1$ and a y-intercept of $-6$ for $x \geq 4$
A line segment with a slope of $-1$ and a y-intercept of $2$ for $x < 4$

Q: What are some real-world applications of the function $y = |x - 4| - 2$ ?

A: The function $y = |x - 4| - 2$ has many real-world applications in fields such as physics, engineering, and economics. Some examples include:

Modeling the motion of an object with a constant acceleration
Describing the behavior of a system with a non-linear response
Analyzing the performance of a financial instrument with a non-linear payoff

Conclusion

In this article, we have answered some frequently asked questions about the function $y = |x - 4| - 2$ . We have covered topics such as the absolute value function, the domain and range of the function, symmetry, maximum and minimum values, and graphing the function. We have also discussed some real-world applications of the function.