What do we want when analyzing a function?

• Domain
• Zeros

• Continuity

  • discontinuities
  • singularities
• End-Behavior

• Behavior

  • intervals where increasing
  • intervals where decreasing
• Global Maximum and Minimum
• Local Maximums and Minimums
• Range
• ...and we would like a nice graph

We want a nice, informative graph

• labeled axes
• intercepts
• endpoints
• vertical asymptotes
• horizontal asymptotes
• arrows

A graph is simply a communication tool. It is inherently inaccurate by the very fact that it is drawn. Algebra and functional reasoning are the only tools for exactness, which is our end goal.

\(\blacktriangleright \) Reasoning: Reasoning is a logical explanation that describes our conclusions, how we arrived at those conclusions, and why we think those conclusions are correct.

Analysis is not a list of conclusions. We are not looking for such a list.

We are certainly not looking for such a list based on viewing a graph.

We are looking for the thought process that arrived at the list of conclusions.

Learning Outcomes

In this section, students will

• analyze functions from their formula.