Mathematics Describing the Real World: Precalculus and Trigonometry Season 1 Episode 4 Rational Functions

The show Mathematics Describing the Real World: Precalculus and Trigonometry season 1, episode 4, titled "Rational Functions," focuses on understanding and graphing rational functions. The episode starts by introducing the concept of rational functions and explaining how they are used to model real-world phenomena such as population growth and drug dosage.

The host starts the show by defining rational functions as functions of the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomials. The host explains how to simplify rational functions and how to find their domains and asymptotes. The episode then shows how to graph rational functions by plotting points and connecting them with lines and curves.

The show provides detailed explanations of how to find the vertical and horizontal asymptotes, how to identify holes, and how to calculate end behavior. The host also demonstrates how to use the rational root theorem to find the roots of a polynomial, which is essential in finding the x-intercepts of a graph.

The episode includes several examples of linear, quadratic, and cubic rational functions, as well as more complex functions involving polynomials with high degrees. The host explains how to use long division or synthetic division to divide polynomials and how to write rational functions in partial fraction form.

The show also introduces the concept of inverse functions and explains how to find the inverse of a rational function. The host provides examples of how to graph inverse functions and how to use the horizontal line test to determine if a function is one-to-one.

Throughout the episode, the host emphasizes the importance of understanding rational functions for real-world applications, such as engineering, physics, and economics. The show demonstrates how rational functions can be used to model the behavior of systems such as mechanical vibrations, electric circuits, and chemical reactions.

The host concludes the episode by summarizing the key concepts covered, including domain, asymptotes, roots, inverse functions, and real-world applications. The episode ends with a challenge problem for the viewers to try.

Overall, Mathematics Describing the Real World: Precalculus and Trigonometry season 1, episode 4, "Rational Functions," provides a comprehensive overview of rational functions and their applications. The show's clear explanations and examples make it easy to understand the material, and the real-world examples make the concepts relevant and engaging.