Polynomial and Rational Functions

Polynomial and Rational Functions

Chapter 5: Introduction and Outcomes

Polynomial and Rational Functions

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.

(From OpenStax College Algebra)

Chapter Sections

  • 5.1 Quadratic Functions
  • 5.2 Power Functions and Polynomial Functions
  • 5.3 Graphs of Polynomial Functions
  • 5.4 Dividing Polynomials
  • 5.5 Zeros of Polynomial Functions
  • 5.6 Rational Functions
  • 5.7 Inverses and Radical Functions
  • 5.8 Modeling Using Variation

OTM Outcomes

1a. Analyze functions. Routine analysis includes discussion of domain, range, zeros, general function behavior (increasing, decreasing, extrema, etc.). In addition to performing rote processes, the student can articulate reasons for choosing a particular process, recognize function families and anticipate behavior, and explain the implementation of a process (e.g., why certain real numbers are excluded from the domain of a given function).*

1b. Convert between different representations of a function.*

1c. Perform operations with functions including addition, subtraction, multiplication, division, composition, and inversion; connect properties of constituent functions to properties of the resultant function; and resolve a function into a sum, difference, product, quotient, and/or composite of functions.*

2a. Recognize function families as they appear in equations and inequalities and choose an appropriate solution methodology for a particular equation or inequality and can communicate reasons for that choice.*

2c. Distinguish between exact and approximate solutions and which solution methodologies result in which kind of solutions.*

2d. Demonstrate an understanding of the correspondence between the solution to an equation, the zero of a function, and the point of intersection of two curves.*

3a. Purposefully create equivalences and indicate where they are valid.*

3b. Recognize opportunities to create equivalencies in order to simplify workflow.*

4a. Interpret the function correspondence and behavior of a given model in terms of the context of the model.*

4c. Determine parameters of a model given the form of the model and data.*

4d. Determine a reasonable applied domain for the model as well as articulate the limitations of the model.*

5a. Anticipate the output from a graphing utility and make adjustments, as needed, in order to efficiently use the technology to solve a problem.*

5d. Use technology and algebra in concert to locate and identify exact solutions.*

6a. Recognize when a result (theorem) is applicable and use the result to make sound logical conclusions and provide counter-examples to conjectures.*

 

Chapter Section Objectives and related OTM standards:

5.1 Quadratic Functions

  • Recognize characteristics of parabolas.(1a)
  • Understand how the graph of a parabola is related to its quadratic function (1a, 1b, 4c)
  • Determine a quadratic function’s minimum or maximum value (1a, 3b)
  • Solve problems involving a quadratic function’s minimum or maximum value (2c, 4a, 4d)

5.2 Power Functions and Polynomial Functions

  • Identify power functions (1a)
  • Identify end behavior of power functions (1a)
  • Identify polynomial functions (1a)
  • Identify the degree and leading coefficient of polynomial functions (1a, 2a, 2d)

5.3 Graphs of Polynomial Functions

  • Recognize characteristics of graphs of polynomial functions (1a)
  • Use factoring to find zeros of polynomial functions (2a)
  • Identify zeros and their multiplicities (1a, 2d)
  • Determine end behavior (1a)
  • Understand the relationship between degree and turning points (1a)
  • Graph polynomial functions (1a, 5a)
  • Use the Intermediate Value Theorem (6a)

5.4 Dividing Polynomials

  • Use long division to divide polynomials.
  • Use synthetic division to divide polynomials.
     

5.5 Zeros of Polynomial Functions

  • Evaluate a polynomial using the Remainder Theorem (3b)
  • Use the Factor Theorem to solve a polynomial equation (2a, 5d, 6a)
  • Use the Rational Zero Theorem to find rational zeros (2a, 5d)
  • Find zeros of a polynomial function (2a, 5d)
  • Use the Linear Factorization Theorem to find polynomials with given zeros (3a, 6a)
  • Use Descartes’ Rule of Signs (6a)
  • Solve real-world applications of polynomial equations (2a, 4a)
     

5.6 Rational Functions

  • Use arrow notation (1a)
  • Solve applied problems involving rational functions (1b, 4a)
  • Find the domains of rational functions (1a)
  • Identify vertical asymptotes (1a)
  • Identify horizontal asymptotes (1a, 2a)
  • Graph rational functions (1a, 2a, 5a)

5.7 Inverses and Radical Functions

  • Find the inverse of an invertible polynomial function (1a, 1c)
  • Restrict the domain to find the inverse of a polynomial function (1a, 1c)

5.8 Modeling Using Variation

  • Solve direct variation problems (1b, 4c, 5d)
  • Solve inverse variation problems (1b, 4c, 5d)
  • Solve problems involving joint variation (1b, 4c, 5d)
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