This section talks about deciphering function information from the graph of a
function, which is not what we want. Our goal is exactness, which can only be
achieved through algebra and functional reasoning.
However, students have extensive experience with graphs, which makes graphs a good
starting point.
Our plan is to use graphs to understand the ideas of function analysis and let that guide our algebraic analysis, which is what we want.
Functions are packages containing three sets (one of which is a set of pairs), which satisfies one rule. That is the struture of a function. In Analysis, the sets are often measurements and the function is how we compare those measurements.
Typical questions we have about functions are:
- When one measurement increases, does the other also increase?
- When one measurement increases, does the other decrease?
- When one measurement changes quickly, does the other change quickly or slowly?
- Which measurements are connected to maximums and minimums in the other measurements?
- If one measurement is smooth, then is the other smooth or does it have sudden changes?
In this section, we will introduce this type of function analysis via graphs. Once we gain a graphical familiarity with this type of analysis, we will extend our thinking to include formulas (Algebra). Our eventual goal is to be exact with our analysis. This will require both our graphical and algebraic tools working together.
Learning Outcomes
In this section, students will...
- communicate via graphs.
- be introduced to counterexamples.
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more examples can be found by following this link
More Examples of Graphical Language