What do we want when analyzing a function?

We would like to analyze all of their characteristics and features.

• 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

In addition, we want algebraic and functional reasoning on how we arrive at our conclusions.

To help us with this goal of algebraic communication, we can use a nice graph. Everything we eventually want to say algebraically corresponds to graphical characteristics.

We don’t want our reasoning to be “look at the graph”, however, we can look at the graph and have it help us translate our reasoning into algebra and function language.

Along with dots representing function pairs, a nice graph, including auxillary graphing items

• intercepts
• endpoints
• vertical asymptotes
• horizontal asymptotes

A graph is a communication tool.

Learning Outcomes

In this section, students will

• analyze functions from their formula.