1. What is an Absolute Value Function?

The absolute value function is a V-shaped graph that represents the distance of a number from zero on the number line. The general form is \( y = a|x - h| + k \), where:

• \( a \) determines the steepness and direction of the V
• \( h \) is the horizontal shift
• \( k \) is the vertical shift

2. How Does the Calculator Work?

The calculator plots the absolute value function:

It calculates 100 points between your specified x-range and connects them to form the graph.

3. Understanding the Graph

Key Features:

• Vertex: The point (h, k) is the "corner" of the V
• Slope: The lines have slopes of ±a
• Symmetry: The graph is symmetric about the line x = h

4. Using the Calculator

Tips:

• Enter the coefficients a, h, and k for your function
• Set appropriate x-min and x-max values to view the relevant portion of the graph
• Positive 'a' values make the V open upward, negative makes it open downward

5. Frequently Asked Questions (FAQ)

Q1: What does the absolute value function represent?
A: It represents distance from zero, always returning non-negative values regardless of input.

Q2: How does changing 'a' affect the graph?
A: Larger |a| makes the V steeper. Negative 'a' flips it upside down.

Q3: What's the difference between |x| and |x - h|?
A: |x| is centered at x=0, while |x - h| shifts the vertex to x=h.

Q4: Can I graph multiple functions at once?
A: This calculator graphs one function at a time for clarity.

Q5: Why is my graph not showing?
A: Check that your x-min is less than x-max and you've entered valid numbers.