Which Graph Represents the Function p(x) = |x – 1|?

Absolute value functions are essential in mathematics, offering Which Graph Represents the Function p(x) = |x – 1|? insights into various real-world phenomena and mathematical principles. When dealing with absolute value functions, understanding their graphical representations is crucial. In this article, we’ll delve into the function $p(x)=|x–1|$ and explore how its graph behaves on the Cartesian plane.

Introduction to Absolute Value Functions

Absolute value functions are mathematical functions that express the distance of a number from zero on the number line Which Graph Represents the Function p(x) = |x – 1|?, disregarding its sign. They are denoted by $|x|$, where $x$ represents any real number. These functions are fundamental in algebra, calculus, and other branches of mathematics.

Understanding the Function $p(x)=|x–1|$

Let’s dissect the function $p(x)=|x–1|$. Here, $x$ is the input variable, and the expression $|x–1|$ represents the absolute value of $x$ minus 1. This function computes the distance between $x$ and 1 on the number line.

Interpreting the Graph of $p(x)=|x–1|$

When we graph $p(x)=|x–1|$, we observe a distinctive V-shaped curve centered at $x=1$. The vertex of this graph occurs at the point (1, 0), representing the minimum value of the function. As $x$ moves away from 1 in either direction, the function’s value increases linearly.

Key Characteristics of the Graph

One key characteristic of this graph is its slope. On the left side of the vertex (for $x<1$), the slope of the graph is -1, indicating a downward slope. On the right side of the vertex (for $x>1$), the slope becomes 1, showing an upward slope.

Graphical Transformation of Absolute Value Functions

Graphical transformations such as shifting, stretching, and reflecting can alter the appearance of absolute value function graphs. For example, shifting the graph of $p(x)=|x|$ horizontally by 1 unit to the right results in the graph of $p(x)=|x–1|$.

Comparing $p(x)=|x–1|$ with Other Absolute Value Functions

Contrasting $p(x)=|x–1|$ with similar functions like $p(x)=|x|$ or $p(x)=|x+1|$ reveals differences in vertex position and slope behavior. Understanding these distinctions aids in graph interpretation and problem-solving.

Real-Life Applications

Absolute value functions have practical applications in various fields such as physics, economics, and engineering. For instance, in physics, these functions can describe the magnitude of a particle’s displacement from a reference point over time Which Graph Represents the Function p(x) = |x – 1|?.

Common Misconceptions

Misconceptions about absolute value functions often arise from misinterpretation of the graph’s behavior, especially concerning vertex location and slope direction. Clearing up these misconceptions enhances understanding and problem-solving skills.

Tips for Graphing Absolute Value Functions

Graphing absolute value functions involves identifying key points such as the vertex, intercepts, and slope. By following step-by-step guidelines and utilizing helpful techniques, one can accurately represent these functions on the Cartesian plane.

Practice Problems

To reinforce understanding, here are a few practice problems:

1. Graph the function $p(x)=|x–1|$ on the Cartesian plane.
2. Determine the vertex and slope of the graph of $p(x)=|x–1|$.
3. Compare and contrast the graphs of $p(x)=|x–1|$ and $p(x)=|x+1|$.

Advanced Concepts

Advanced topics related to absolute value functions include transformations involving multiple Which Graph Represents the Function p(x) = |x – 1|? absolute value functions, piecewise functions, and applications in optimization and modeling.

Online Resources and Tools

Numerous online resources and software tools are available for graphing absolute value functions, including Desmos, GeoGebra, and Wolfram Alpha. These resources offer interactive features and tutorials for additional learning and practice.

Seeking Help and Support

If you encounter difficulties understanding absolute value functions or graphing techniques, don’t hesitate to seek help from teachers, tutors, or online communities dedicated to mathematics education.

Conclusion

Understanding the graph of $p(x)=|x–1|$ and other absolute value functions is essential for mastering algebraic concepts and problem-solving skills. By grasping the key characteristics and behaviors of these functions, you’ll be better equipped to tackle various mathematical challenges.

FAQs

1. What does the absolute value function $|x–1|$ represent? The function represents the distance between the input variable $x$ and the number 1 on the number line.
2. How do you find the vertex of the graph of $p(x)=|x–1|$? The vertex occurs at the point where $x=1$, resulting in the coordinates (1, 0).
3. What is the significance of the slope of the graph of $p(x)=|x–1|$? The slope indicates the rate of change of the function’s value as $x$ moves away from 1 on the number line.
4. Can absolute value functions have more than one vertex? Yes, absolute value functions can have multiple vertices if they undergo transformations such as shifting or stretching.
5. Are there any real-world applications of absolute value functions? Yes, absolute value functions are used in various real-world scenarios, including physics (to describe displacement), economics (for optimization problems), and engineering (in signal processing).