Introduction to Periodic Functions
Life is dense with phenomena that repeat in regular intervals. Throughout history, many Indigenous peoples have used this regularity to build cultural narratives and direct key activities, such as agriculture, hunting, and fishing. Semi-nomadic aboriginal people dwelling in the tall pines of East Texas as early as 200 B.C. called the Caddo Tribe lived around the rivers and marshes in the area. They used the regularity of seasonal rainfall and change in water level to determine the best times to fish, hunt, and grow crops. The knowledge of how to predict seasonal changes necessary for the survival of the tribe was passed down orally from elders and did not exist in a book.
Did you know that the intersection of mathematics and culture is a field of study called Ethnomathematics? The Institue of Texan Cultures had a Math + Culture Tour developed by the UTSA Department of Mathematics which highlighted the many ways mathematics was used but not necessarily documented in various cultures historically. In fact, many of the skills required by settlers in west Texas involved mathematical thinking and problem solving. For instance, the construction of a house should be in a location with enough shade for the hot summers but enough light to warm sufficiently in cold winters. The roof should be built to catch valuable rainwater for use in the garden given the construction materials available (why don’t adobe homes have a steep pitch?). Many of the decisions made by the individuals in a community determined the overall success and survival of the whole group.
Eventually, civilizations recorded these developments and western mathematicians created mathematical models in the form of functions to track and predict fluctuations in nature such as the climate and tides. In more formal terms, a function that repeats its values in regular intervals is known as a periodic function. The graphs of such functions show a general shape reflective of a pattern that keeps repeating. This means the graph of the function has the same output at exactly the same place in every cycle. And this translates to all the cycles of the function having exactly the same length. So, if we know all the details of one full cycle of a true periodic function, then we know the state of the function’s outputs at all times, future and past.
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In Chapter 3, we will investigate various examples of periodic functions. You will explore sine and cosine functions and identify properties such as amplitude, period, and phase shift of a given sinusoidal. You will have the opportunity to analyze graphs showing other trigonometric functions and formulate the equation of the function when given the graph of a sinusoidal and identify their asymptotes.
A complete list of the Key Takeaways from the chapter is organized in the table below:
Key Takeaways
- Graph the function [latex]f(x)=\sin(x)[/latex] and [latex]g(x) =\cos(x)[/latex] on a cartesian coordinate plane using key points and identify the function’s domain and range.
- Determine the Amplitude, Period, and Phase Shift of a sinusoidal function given its equation.
- Find equations of sinusoidal functions given their equations.
- Graph the basic trigonometric functions using key points, vertical asymptotes, and reciprocal identities, as needed.
