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Graphing Absolute Value Functions

 Teaching the Mathematical Practices

6 Communicate Precisely  Encourage students to routinely write or explain their solution methods. Point out that they should use clear definitions when they discuss their solutions with others.

The absolute value function is a type of piecewise-linear function. Because the absolute value of a number is always nonnegative, all the values in the range of the parent function, f ( x ) = x ,  are also nonnegative.
An absolute value function is written as f ( x ) = a x - h + k ,  where a, h, and k are constants and f ( x ) ≥ 0  for all values of x.
The vertex is either the lowest or highest point of a function. For the parent function, y = x ,  the vertex is at the origin.

Key Concept: Absolute Value Function
 What Students Are Learning

An absolute value function is a piecewise-defined function, whose graph consists of two pieces that form a V. The vertex of the graph of the parent function is the origin, and the y-values of all points on the graph are nonnegative. The graph is symmetric with respect to the y-axis, as the absolute value of a number is the same as the absolute value of its opposite.

 Think About It!

 Answer

Sample answer: The graph is symmetric about the y-axis. Part of the line  f ( x ) = x  is reflected across the y-axis to make the other part of the graph.

 Common Misconception

Some students may think that the graph of any absolute value function will lie completely above the x-axis. Explain that just as with other functions, transformations of the function will relocate the graph, and the resulting graph may, in fact, contain points that lie below the x-axis.