3: Polynomial and Rational Functions
- Page ID
- 203401
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35-mm film, once the standard for capturing photographic images, has been made largely obsolete by digital photography. (credit “film”: modification of work by Horia Varlan; credit “memory cards”: modification of work by Paul Hudson)
Digital photography has dramatically changed the nature of photography. No longer is an image etched in the emulsion on a roll of film. Instead, nearly every aspect of recording and manipulating images is now governed by Mathematics. An image becomes a series of numbers, representing the characteristics of light striking an image sensor. When we open an image file, software on a camera or computer interprets the numbers and converts them to a visual image. Photo editing software uses complex polynomials to transform images, allowing us to manipulate the image in order to crop details, change the color palette, and add special effects. Inverse functions make it possible to convert from one file format to another. In this chapter, we will learn about these concepts and discover how Mathematics can be used in such applications.
- 3.1: Quadratic Functions
- This section covers quadratic functions, focusing on their general and standard (vertex) forms. It explains how to find and interpret key features such as the vertex, axis of symmetry, and zeros. It also demonstrates graphing parabolas and solving quadratic equations using the Quadratic Formula, completing the square, and factoring. Advanced topics include solving absolute value equations involving quadratic expressions and graphing the absolute value of a quadratic function.
- 3.2: Power Functions and Polynomial Functions
- This section discusses power and polynomial functions, focusing on their definitions, properties, and graphs. It explains the general form of polynomial functions, the significance of the leading term and degree, and how these influence end behavior. Examples demonstrate graphing polynomial functions and analyzing their key features, including turning points and continuity.
- 3.3: Graphs of Polynomial Functions
- This section explores the graphs of polynomial functions, focusing on key characteristics such as end behavior, intercepts, and turning points. It discusses how the degree and leading coefficient influence the graph's shape and analyzes multiplicity's role in determining the behavior of the graph at roots. Examples illustrate how to sketch and interpret polynomial graphs effectively.
- 3.4: Dividing Polynomials
- This section covers methods for dividing polynomials, including long division and synthetic division. It explains how to use these techniques to divide a polynomial by a linear or higher-degree polynomial, interpret the results, and find remainders. Examples illustrate each method step-by-step, helping to solve polynomial division problems efficiently.
- 3.5: Rational Functions
- This section introduces rational functions, exploring their key features such as domain, vertical and horizontal or slant asymptotes, and intercepts. It discusses how to analyze and graph rational functions, focusing on behavior near asymptotes and at infinity. Examples demonstrate the steps to identify these features and interpret their implications for the function's graph.
- 3.6: Revisiting Inverses
- This section discusses the inverses of polynomial and rational functions, focusing on determining whether a function has an inverse, finding the inverse algebraically, and understanding the domain and range restrictions necessary for invertibility. It provides examples to demonstrate these concepts and explores graphical relationships between functions and their inverses, including symmetry about the line \(y = x\).