How to Graph Absolute Value Functions

Graphing absolute value functions involves understanding how the absolute value transformation affects the original function. The absolute value of a number is its distance from zero, meaning it's always non-negative. This characteristic shapes the distinctive V-shaped graph.

Understanding Absolute Value

The absolute value function, denoted as |x|, is defined as:

• |x| = x, if x ≥ 0
• |x| = -x, if x < 0

This means any negative value inside the absolute value is made positive, while positive values remain unchanged.

Steps to Graph an Absolute Value Function

Let's consider the general form: y = a|x - h| + k

Here's how to graph it:

1. Identify the Vertex: The vertex is the "corner" of the V-shaped graph and is located at the point (h, k). In y = a|x - h| + k, h shifts the graph horizontally (opposite of the sign, so x-2 shifts the graph right 2 units), and k shifts the graph vertically.

2. Determine the Direction of Opening:

   • If a > 0, the graph opens upwards (a standard "V").
   • If a < 0, the graph opens downwards (an inverted "V"). The graph is reflected across the x-axis.

3. Find Additional Points: Choose x-values on both sides of the vertex and calculate the corresponding y-values. Two points are usually enough, but more points provide better accuracy.

4. Plot the Points and Draw the Graph: Plot the vertex and the additional points. Connect them with straight lines to form the V-shape.

Example

Let's graph y = |x - 2| + 1:

1. Vertex: The vertex is at (2, 1) because h = 2 and k = 1.

2. Direction: Since the coefficient of the absolute value term is 1 (positive), the graph opens upwards.

3. Additional Points:

   • If x = 0, y = |0 - 2| + 1 = | -2 | + 1 = 2 + 1 = 3. So, the point (0, 3) is on the graph.
   • If x = 4, y = |4 - 2| + 1 = |2| + 1 = 2 + 1 = 3. So, the point (4, 3) is on the graph.

4. Plot and Draw: Plot the vertex (2, 1) and the points (0, 3) and (4, 3). Connect the points to form the V-shape.

Effects of a, h, and k

• a:

  • |a| > 1: The graph is vertically stretched (narrower).
  • 0 < |a| < 1: The graph is vertically compressed (wider).
  • a < 0: The graph is reflected over the x-axis.

• h: Shifts the graph horizontally.

  • h > 0: Shifts the graph h units to the right.
  • h < 0: Shifts the graph h units to the left.

• k: Shifts the graph vertically.

  • k > 0: Shifts the graph k units up.
  • k < 0: Shifts the graph k units down.

Summary

Graphing absolute value functions is straightforward. Identify the vertex, determine the direction of opening, find a couple of additional points, and connect them to form the distinctive V-shape. Understanding how a, h, and k affect the graph allows for quick sketching and accurate representation of these functions.