Crossroads in Mathematics

Example 13

A Pure Mathematics Problem (Absolute Value Functions)

Thus far, most of the example problems have had a context, a story line. Some of the problems are obviously contrived (e.g., the dart board problem, 12), while others are based on genuine data (e.g., the planet problem, 13). Mathematics presented in context is often more attractive to students and will engage their interest and attention to a greater degree than mathematics without a context.

Lest there be any doubt, however, there are many worthwhile and important pure mathematics problems, like the one presented below. This problem presents pure mathematics in a way that allows for group interaction and use of technology to develop critical concepts.

Part A. Sketch a graph for each function. On the same axes, in a different color, dash in the graph produced by omitting the absolute value operation. Check your graphs with a graphing utility.

1. 2.
3. 4.
5. 6.

Conclusions: Describe how applying the absolute value operation before the characteristic operation affects the graph.

Part B. Sketch a graph for each function. On the same axes, in a different color, dash in the graph produced by omitting the absolute value operation. Check your graphs with a graphing utility.

1. 2.
3. 4.
5. 6.

Conclusions: Describe how applying the absolute value operation after the characteristic operation affects the graph.

Part C. Sketch a graph for each function. On the same axes, in a different color, dash in the graph produced by omitting the absolute value operation.

1. 2.
3. 4.

Conclusions: Do your descriptions from part A and part B explain how to produce the graphs involving absolute value from their standard graphs? Give steps to produce such a graph.

Part D. Generalize your conclusions. Use the steps you listed in part C to sketch the graph of each of the following functions. Don't use any other method. Now by checking points or using your grapher determine if the graph is correct.

1. 2.
3. 4.

Conclusions: If your steps did not cover these cases, explain why. Redo your descriptions to cover these more general cases.