Great developments were made from the time of Pythagoras about 400 BC until the fall of the Roman Empire about 400 AD. Then, for 1000 years very little new math was developed. About 1550, an Italian mathematician by the name of Cardano derived an algebraic method for finding the roots of a cubic function. It was not until over a hundred years later when calculus was developed by Newton that the turning points and point of inflection could be found.
Even so, Cardano’s method for finding the roots of a cubic are not taught in either algebra or calculus classes today because of its difficulty. We can with relative simplicity use calculus to find the turning points and point of inflection. There is, however, another cubic formula that is little known today and not taught in algebra or calculus courses nor is it found in textbooks. It is a mystery as to why.
This formula, which is presented in this Spidercard, is simple and finds the turning points and points of inflection and other important characteristics of a cubic function using algebra and not calculus. It is the last algebraic method for finding the turning points and point of inflection of polynomial functions. Functions higher than degree three require calculus.
In addition, this Spidercard shows how the cubic formula can also describe certain characteristics of the antiderivative and derivatives of a cubic function and serves as a bridge to both differential and integral calculus. Lastly, the card introduces you to one of the “forgotten points” namely, a “point of jerk.” You will be amazed at what this simple formula tells us about a cubic function without calculus.
Jerk And Jounce - The Forgotten Points
There are many different points that can be plotted on the curve of a function. Intercepts, Turning Points, and Points of Inflection are most commonly discussed in classes and textbooks.
⦁The y-intercept is where a function crosses the vertical axis. ⦁The x-intercepts are where a function crosses or touches the horizontal axis. ⦁Turning Points are where a function changes direction (up to down or down to up). ⦁Points of Inflection are where a function changes concavity (bending up to bending down or down to up).
Each point on a real-life function has a specific meaning and importance. The graph below is that of the growth in the number of COVID cases based on data reported during the early weeks of the pandemic. It was this graph that the health experts referred to during their daily briefings since early 2020. The middle point of the five points in the graph on the left is a point of inflection and is the point most often alluded to. The turning point on the graph to the right correlates directly with the point of inflection on the graph to the right. Both points were referred to during the daily health briefings.
The other four points on the left are as important if not more important yet were never discussed. These are the “forgotten points” that are the subject of this Spidercard. The forgotten points are not discussed in college classes or nor found in textbooks, although they should be. They are virtually unknown but have important relevance in virtually every career from business to psychology to engineering. They are not difficult to find nor to interpret their importance to real world applications. This Spidercard illustrates for the first time the hierarchy of functions and related hierarchy of points so essential to a complete function analysis of real-world applications. This card shows how these two forgotten points relate to the intercepts, inflection points, and turning points and should be integrated into textbooks and advanced algebra and calculus courses.
We observe the real world and collect data. We then describe it by developing a mathematical relation using statistical regression that attempts to explain the real world accurately enough for us to make rational decisions. The most recent example is COVID 19 where we were able to describe the growth of COVID cases by means of an S-shaped logistic function that was shown on television as our medical experts kept us informed.
Often the real-world is described by polynomials, exponential, logarithmic, rational, and other types of functions first learned in algebra and then analyzed using calculus and other mathematical methods. Rational decisions are then based on the analysis of the real-world function.
The Function Analysis Spidercard describes the complete set of the important characteristics of interest such as turning points, concavity, and the rest of the important characteristics that are what decision-makers use to base their decisions and also use to inform those of us that live in the real-world. These characteristics also become the subject of conversations we use in our daily life as we bop along the boardwalk, at neighborhood barbeques, and at the office water cooler.
This Spidercard then is important not only for students but also professionals in their respective careers, and the general population who need to understand what they see in newspapers and on social platforms, hear on TV, or are part of the general conversation at the office or other social gatherings.