analysis

We want precise analysis. We want to be exact.

The only way to do this is by using algebra. Graphs are inherently inaccurate and there is nothing we can do about this.

Our goal is rigorous algebraic reasoning.

However, we are just starting out. We don’t have all of our algebraic tools yet. That is what this course is introducing. So, sometimes we have to provide graphical reasoning. It will be approximate rather than precise. We’ll just have to understand this. When we can be precise with algebra, then that is what we want.

We want precise analysis, which comes through logical reasoning and algebraic calculations, when we can get it.

We want to explain how we know we are exactly correct, but will settle for graphical feelings of correct, if that is all we can get.

Our analysis will slowly shift from graphical to algebraic.

To this end, we need to sort out our vocabulary.

Function Words

We are describing functions characteristics. These are the words we use for functions.

\[ \begin{array}{ll} \text {domain} & \text {local maximum} \\ \text {zeros} & \text {local minimum} \\ \text {continuity} & \text {range} \\ \text {discontinuity} & \text {value} \\ \text {singularity} & \text {image} \\ \text {end-behavior} & \text {preimage} \\ \text {behavior} & \text {range partner} \\ \text {increasing} & \text {onto} \\ \text {decreasing} & \text {one-to-one} \\ \text {global maximum} & \text {well-defined} \\ \text {global minimum} & \text {one-to-one} \end{array} \]

Graph Words

We are describing functions characteristics. We use graphs to visually describe these characteristics. We decipher the visual information from the graph and make conclusions about the function.

When talking about a graph, we use these words.

\[ \begin{array}{ll} \text {point} & \text {coordinate} \\ \text {uphill} & \text {axis} \\ \text {downhill} & \text {intercept} \\ \text {slope} & \text {vertical asymptote} \\ \text {singularity} & \text {horizontal asymptote} \\ \text {break} & \text {jump} \\ \text {hole} & \text {arrow} \\ \text {highest} & \text {lowest} \\ \text {hill} & \text {valley} \\ \text {solid} & \text {hollow} \\ \text {distance} & \text {above} \\ \text {below} & \text {vertex} \end{array} \]

For instance, functions don’t have asymptotes. Their grahps do. We use asymptotes to visually encode information about a function’s singularities or end-behavior.

When analyzing a function, we are trying to get away from these words.

When analyzing a function, we want to use algebraic words.

Algebraic Words

We are describing functions characteristics. We use formulas and equations to algebraically encode information about a function. We then use our algebraic knowledge to decipher function information.

When analyzing a function, we want to use function words and algebra words.

\[ \begin{array}{ll} \text {formula} & \text {equation} \\ \text {coefficient} & \text {leading coefficient} \\ \text {term} & \text {leading term} \\ \text {numerator} & \text {denominator} \\ \text {base} & \text {fraction bar} \\ \text {exponent} & \text {inside} \\ \text {log} & \text {outside} \\ \text {variable} & \text {constant term} \\ \text {root} & \text {constant multiple]} \\ \text {radical} & \text {absolute value bars} \\ \text {parentheses} & \text {sum} \\ \text {difference} & \text {product} \\ \text {quotient} & \text {fraction bar} \\ \text {subscript} & \text {superscript} \\ \text {expession} & \text {index} \\ \text {solution} & \text {notation} \end{array} \]

Our goal is algebraic reasoning function analysis.

Our algebra is not strong enough to do this all of the time.

Learning when the algebra will not produce the reasoning we want is part of learning mathematics.

Algebra and function reasoning first.

Then graphical reasoning.

But, always some reasoning. Always an explanation. Not just a declaration of facts.

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more examples can be found by following this link
More Examples of Visual Behavior

2026-09-20 14:21:28