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How to Graph an Absolute Value Function?

Published in Graphing Functions 3 mins read

To graph an absolute value function, identify the vertex first, choose x-values to its left and right, calculate the corresponding y-values, plot the points, and draw the V-shaped graph.

Here's a step-by-step guide on how to graph an absolute value function:

Steps to Graphing an Absolute Value Function

  1. Identify the Vertex: The vertex is the point where the absolute value function changes direction. For a function in the form f(x) = a|x - h| + k, the vertex is at the point (h, k). The value of 'a' determines whether the graph opens upward (a > 0) or downward (a < 0).

  2. Create a Table of Values: Choose an x-value for the vertex. Then select two x-values that are less than the x-value of the vertex, and two x-values that are greater than the x-value of the vertex. Calculate the corresponding y-values for each chosen x-value using the absolute value function.

  3. Plot the Points: Plot the ordered pairs (x, y) you calculated in the table of values on a coordinate plane.

  4. Draw the Graph: Connect the plotted points. The graph of an absolute value function is V-shaped. Draw straight lines from the vertex outwards in both directions, passing through the plotted points. The lines should extend infinitely in both directions, indicating that the function continues beyond the plotted points.

Example

Let's graph the absolute value function f(x) = |x - 2| + 1.

  1. Vertex: The vertex is at (2, 1).

  2. Table of Values:

    x Calculation y = f(x)
    0 0 - 2
    1 1 - 2
    2 2 - 2
    3 3 - 2
    4 4 - 2

  3. Plot the Points: Plot the points (0, 3), (1, 2), (2, 1), (3, 2), and (4, 3) on a coordinate plane.

  4. Draw the Graph: Connect the points to form a V-shaped graph with the vertex at (2, 1).

Key Considerations:

  • The absolute value function always returns a non-negative value.
  • The graph is symmetric around a vertical line that passes through the vertex. This line is called the axis of symmetry.
  • The "a" value in f(x) = a|x - h| + k stretches or compresses the graph vertically. If |a| > 1, the graph is stretched vertically; if 0 < |a| < 1, the graph is compressed vertically. If a < 0, the graph is reflected across the x-axis.

By following these steps, you can accurately graph any absolute value function.

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