Graphing Absolute Value Functions: A Comprehensive Guide with Worked Examples
This guide provides a comprehensive walkthrough of graphing absolute value functions, complete with worked examples to solidify your understanding. We'll cover the key concepts, techniques, and common pitfalls, equipping you to tackle any absolute value graphing problem with confidence. Whether you're a student working through a worksheet or a math enthusiast looking to sharpen your skills, this guide is for you.
Understanding the Absolute Value Function
The absolute value of a number is its distance from zero. Therefore, it's always non-negative. The absolute value function, denoted as |x|, is defined as:
- |x| = x, if x ≥ 0
- |x| = -x, if x < 0
This seemingly simple definition leads to a unique V-shaped graph. The vertex of the "V" is located at the point where the expression inside the absolute value becomes zero.
Graphing Basic Absolute Value Functions
Let's start with the simplest form: f(x) = |x|. Its graph is a V-shaped curve with the vertex at (0, 0). The slope is -1 for x < 0 and 1 for x > 0.
Transformations of Absolute Value Functions
More complex absolute value functions involve transformations of the basic |x| graph. These transformations include:
-
Vertical Shifts: Adding a constant k to the function shifts the graph vertically. f(x) = |x| + k shifts the graph k units upward (if k > 0) or downward (if k < 0).
-
Horizontal Shifts: Adding a constant h inside the absolute value shifts the graph horizontally. f(x) = |x - h| shifts the graph h units to the right (if h > 0) or to the left (if h < 0).
-
Vertical Stretches/Compressions: Multiplying the function by a constant a stretches or compresses the graph vertically. f(x) = a|x| stretches the graph vertically if |a| > 1 and compresses it if 0 < |a| < 1. If a is negative, the graph is reflected across the x-axis.
-
Horizontal Stretches/Compressions: Similar to vertical stretches, a constant b within the absolute value affects horizontal stretching/compressing. The function f(x) = |bx| compresses the graph horizontally if |b| > 1 and stretches it if 0 < |b| < 1. A negative value for b reflects the graph across the y-axis.
Worked Examples
Let's illustrate these transformations with examples:
Example 1: Graph f(x) = |x - 2| + 1
This graph is a shift of the basic |x| graph: 2 units to the right and 1 unit upward. The vertex is at (2, 1).
Example 2: Graph f(x) = -2|x + 3|
This graph involves a reflection across the x-axis (due to the negative sign), a vertical stretch by a factor of 2, and a horizontal shift of 3 units to the left. The vertex is at (-3, 0).
Example 3: Graph f(x) = 1/2|x| - 3
This graph involves a vertical compression by a factor of 1/2 and a vertical shift of 3 units downward. The vertex is at (0, -3).
Solving Problems from Worksheets
When tackling problems from a worksheet, systematically identify the transformations applied to the basic |x| function. This approach will help you accurately determine the vertex and the shape of the graph. Remember to pay close attention to the signs and coefficients to correctly interpret the transformations.
Conclusion
Graphing absolute value functions becomes straightforward once you understand the basic graph and the effect of transformations. By practicing with various examples and using the techniques outlined above, you'll develop proficiency in graphing these functions and accurately interpreting their properties. Remember to always identify the vertex first—this is the cornerstone of accurately graphing absolute value functions.
