What do we want when analyzing a function?

$\blacktriangleright$ First, and foremost, we want an algebraic description of every detail about our function.

• domain
• zeros
• discontinuities and singularities
• intervals of continuity
• critical numbers
• intervals where increasing and decreasing
• global maximum and minimums
• local maximums and minimums
• end-behavior
• limiting behavior

$\blacktriangleright$ Secondly, we would like a nice graph, including corresponding important points and auxilary graphing items.

• intercepts
• endpoints
• vertical asymptotes
• horizontal asymptotes

We often turn to technology for a nice graph and we must remember that it is plotting individual points and may not understand the larger implications. As humans, we possess more information from our algebraic description and can enhance the graph with exaggerations and auxilary graphing items to help the reader understand the function better.

Remember: We are not using our graph as reasoning. We are not drawing conclusions and pointing to the graph and saying ”because of the graph”. We are using the graph to help our algebraic and functional reasoning.

Learning Outcomes

In this section, students will

• analyze functions from their formula.