Graph Of Absolute Value Of X 1

Graphing the Absolute Value of x - 1: A Comprehensive Guide

The absolute value function, denoted as |x|, represents the distance of a number 'x' from zero on the number line. It's a fundamental concept in mathematics with numerous applications across various fields. Understanding the graph of the absolute value function, specifically y = |x - 1|, is crucial for comprehending transformations, solving inequalities, and working with more complex functions. This article provides a comprehensive guide to graphing y = |x - 1|, exploring its key features, transformations, and applications.

Understanding the Absolute Value Function

Before diving into the specifics of y = |x - 1|, let's review the core characteristics of the absolute value function, y = |x|.

• Definition: The absolute value of a number x, denoted as |x|, is its distance from zero. This means |x| is always non-negative.

  • If x ≥ 0, then |x| = x.
  • If x < 0, then |x| = -x.

• Graph: The graph of y = |x| is a V-shaped graph with its vertex at the origin (0,0). The left branch is a reflection of the right branch across the y-axis. The slope of the right branch is +1, and the slope of the left branch is -1.

• Key Features:

  • Vertex: (0,0)
  • Domain: All real numbers (-∞, ∞)
  • Range: All non-negative real numbers [0, ∞)
  • Symmetry: Symmetric about the y-axis (even function).

Graphing y = |x - 1|

The equation y = |x - 1| represents a transformation of the basic absolute value function, y = |x|. Specifically, it involves a horizontal shift to the right by one unit.

To graph y = |x - 1|, we can follow these steps:

1. Identify the Transformation: The expression inside the absolute value, (x - 1), indicates a horizontal shift. Remember that a transformation of the form f(x - c) shifts the graph of f(x) to the right by c units. Therefore, y = |x - 1| shifts the graph of y = |x| one unit to the right.

2. Find the Vertex: The vertex of y = |x| is (0, 0). Since we shifted the graph one unit to the right, the vertex of y = |x - 1| is (1, 0).

3. Plot Key Points: We can plot additional points by substituting values of x into the equation:

   • If x = 0, y = |0 - 1| = 1. Point (0, 1)
   • If x = 1, y = |1 - 1| = 0. Point (1, 0) (Vertex)
   • If x = 2, y = |2 - 1| = 1. Point (2, 1)
   • If x = -1, y = |-1 - 1| = 2. Point (-1, 2)
   • If x = 3, y = |3 - 1| = 2. Point (3, 2)

4. Sketch the Graph: Connect the plotted points to create the V-shaped graph. The graph will be identical to the graph of y = |x|, but shifted one unit to the right.

Key Characteristics of y = |x - 1|

• Vertex: (1, 0)
• Domain: All real numbers (-∞, ∞)
• Range: All non-negative real numbers [0, ∞)
• x-intercept: (1, 0)
• y-intercept: (0, 1)
• Symmetry: The graph is not symmetric about the y-axis, unlike y = |x|. It's symmetric about the vertical line x = 1.

Solving Inequalities Involving Absolute Value

Understanding the graph of y = |x - 1| is essential for solving inequalities involving absolute value. For instance, consider the inequality |x - 1| ≤ 2.

Graphically, this inequality represents the region where the graph of y = |x - 1| is below or on the line y = 2. Solving this graphically, we find the x-values where the graph intersects y = 2. This happens at x = -1 and x = 3. Therefore, the solution to the inequality is -1 ≤ x ≤ 3.

Applications of Absolute Value Functions

Absolute value functions have widespread applications in various fields:

• Physics: Calculating distances and magnitudes of vectors.
• Engineering: Modeling error tolerances and deviations from ideal values.
• Computer Science: Implementing algorithms involving distances and comparisons.
• Economics: Analyzing deviations from a mean or target value.
• Data Analysis: Calculating deviations and errors in data sets.

Comparing y = |x| and y = |x - 1|

The key difference between y = |x| and y = |x - 1| lies in the horizontal shift. y = |x - 1| is essentially the graph of y = |x| translated one unit to the right. This shift affects the x-intercept, y-intercept, and the line of symmetry.

Advanced Transformations: Exploring Variations

While we've focused on y = |x - 1|, let's explore other transformations:

• y = |x + 1|: This shifts the graph of y = |x| one unit to the left.
• y = |x| + 1: This shifts the graph of y = |x| one unit up.
• y = |x| - 1: This shifts the graph of y = |x| one unit down.
• y = 2|x - 1|: This stretches the graph of y = |x - 1| vertically by a factor of 2.
• y = (1/2)|x - 1|: This compresses the graph of y = |x - 1| vertically by a factor of 1/2.
• y = -|x - 1|: This reflects the graph of y = |x - 1| across the x-axis.

Understanding these transformations is crucial for graphing more complex absolute value functions and solving related problems.

Piecewise Definition and its Implications

The absolute value function can be defined piecewise:

f(x) = |x - 1| =  
  x - 1, if x ≥ 1
  -(x - 1), if x < 1

This piecewise definition clarifies how the function behaves for different intervals of x. For x ≥ 1, the function is a simple linear function with a slope of 1. For x < 1, it's a linear function with a slope of -1. This piecewise definition is useful for evaluating the function at specific points and for solving inequalities.

Solving Equations Involving Absolute Value

Solving equations involving absolute values often involves considering two cases based on the piecewise definition. For instance, to solve |x - 1| = 2, we solve two separate equations:

• x - 1 = 2 => x = 3
• -(x - 1) = 2 => x = -1

Therefore, the solutions are x = 3 and x = -1. This aligns with the graphical solution where the horizontal line y = 2 intersects the graph of y = |x - 1| at x = 3 and x = -1.

Conclusion

The graph of y = |x - 1| provides a fundamental example of how transformations affect the basic absolute value function. By understanding its key features, transformations, and applications, we can effectively utilize absolute value functions to solve problems across various mathematical and real-world contexts. Mastering the graphing of y = |x - 1| builds a strong foundation for tackling more complex absolute value functions and their related applications. Remember that practice is key to developing a robust understanding of these concepts. Experiment with different transformations, solve various equations and inequalities, and explore the practical applications to solidify your knowledge.