Unit 4: Polynomial and Rational Functions

Section 4.1

Video 1: Linear functions

Video 2: The basic idea of linear regression

Video 3: Linear Depreciation

Video 4: Marginal cost

Section 4.2

Read section 4.2: Read p.229 to 331.

You will not be tested on this, but it is a nice way to see why we are studying the stuff in the upcoming sections.

Video 5: Graphing quadratic functions using completing the square and translations.

If you are not confident with completing the square, click on the link or review the idea by reading your textbook.

Review of completing the square.

Video 6: Graphing quadratics that are “wide” and “skinny”.

Finding maxima and minima of functions that are related to quadratics.

See the end of video 5 from this section.

Video 7: The importance of optimization (finding maxima and minima)

Video 8: Example involving a function related to a quadratic

Video 9: Another example involving a function related to a quadratic

Section 4.4 Defining functions of things “in terms of” other things.

Video 10: Introduction and examples

Video 11: A more involved example

    • p.266: #1,3,4,6,7,9,13,15,18,21,33,35,41,44,51

Honors: You have two options this time. See the details below.

  • Everyone will do this part: p.244: Mini project 1.
  • Option  1: p. 244: Mini project 2. Variation: You will write a program that finds the area under the year vs. oil-consumed curve. This area is the total oil consumed at any given year. You will be able to compare your program’s predictions to those of the equations that the authors provide. It might help to read the first few paragraphs of this article about integration.  Your program with approximate the area under the curve by dividing it into many rectangles whose areas can be calculated and summing those areas. We will, of course, discuss this idea much more thoroughly in class. Possible bonus idea: Use matplotlib to generate graphs that show what your numerical integration function is doing. This could possibly count toward a future honors project, or be exchanged for problems that you find to be “busy work”.
  • Option 2: Page 284. Complete this mini-project (note that the directions for part 2 are the same as for part 1). As part of this project, you will turn in graphs that were generated using python’s matplotlib. If you are feeling very ambitious, you might check out the Coursera course “Machine Learning” which uses the idea of fitting lines to data as a launching point for writing far more complicated programs that make computers learn to do things as well as and often better than we humans! If you find the idea compelling, consider this as a strong candidate for a senior project next year. It would be even better if a couple of you chose to work together on it.

Section 4.6

Video 13: Definition of Polynomial function, power function and degree of polynomial.

Video 14: End behavior Part I

Video 15: End behavior Part II

Video 16: Turning points and approximate behavior near roots.

Assignment 4.6:

  • p.299: # 1, 2, 3, 7, 9, 13,17,18,19,20,21,22,23,24,27,29,31,47-50,53

Section 4.7

Video 17: Rational functions Introduction

Video 18: Translations from long-division and Horizontal asymptotes from factorization tricks (Limits…kind of.)

Assignment 4.7:

  • p.313: # 1-13 odd, 19,23,25,27,35,37,48