Chapter 1 Notes for Math 176
Precalculus with Limits, Second Edition

In order to tie comments to specific locations in the book, I have used the available page ruler sheet to identify lines in the text. A copy of that page ruler sheet can be printed from The Index Sheet.

Chapter 2: Polynomials and Rational Functions
2.1 Quadratic Functions and Models
Page	Line #	Notes
129	9	The Standard Form of a Quadratic. The form f(x) = a(x-h)2 + k, a 0 is, as we have noted before shorthand for f = { (x,y) | y = a(x-h)2 + k, a 0 } which we can easily rewrite as f = { (x,y) | y - k = a(x-h)2, a 0 } giving rise to the concept that the vertex will be at the point (h,k).
		The book does not cover "conic sections" until chapter 10, and it is not part of the 176 course. However, the other approaches to to the parabloa as a conic section are important and probably should be introduced here. This is especially true in terms of giving the definition of a parabloa as a set of points equidistant from a given point and line. A parabola of the form y = ax² + bx + c is a special version of that definition where the line is restriced to be a horizontal line. In particular we should get to the form ( x - h )² = 4 p (y - k) in comparision to the "other" standard form for the parabola.
131	3	The book gives the special calculations for findng the minimum or maximum (depending if the parabola opens up or down, respectively) values of a paraola and the location, the x value, at that minimum or maximum. I would sure hate to have to memorize these, especially given our ability to find them, at least numerically, via the functions on the calculator. On top of that, we could write a program to do this. (We may write such a program in class.)
134	18	Problems like #81 are among the most common supposedly "real life" problems that we use to test people on quadratices, and thus on parabolas. Be sure that you understand how to set up such problems.
2.2 Polynomial Functions of Higher Degree
Page	Line #	Notes
138	3	The "Leading Coefficient Test" is really a great introduction to just looking at functions, any numeric functions, in special situations. Thus, what we are really saying is that without regard to the size of the coefficients in the quadratic equation, the x² term will dominate the entrre functiobn for values of x that are either extremely negative or extremely positive. We come back to similar approaches later in the chapter when we look at asymptotes.
139	32	Knowing the number of zeros and the number of turning points is helpfuul in graphing the function. Finding the zeros, the roots, of the polynomial is a big challenge. Clearly, our calculators can do great things with this. ou will want to be completely comfortable witht eh TABLE feature on the calculator. Be sure to ask if there are any issues using this feature.
143	3	The Intermediate Value Theorem is important. You want to be sure that it makes sense and that the more formal definition, given in the middle of page 143, is clear.
147	11	Problems such as #97 and #98 are more "favorite" questions that are supposed to be real world examples of using quadratics.
2.3. Polynomialss and Synthetic Division
Page	Line #	Notes
2	36
2.4 Complex Numbers
Page	Line #	Notes
2	36
2.5 Zeros of Polynomial Functions
Page	Line #	Notes
2	36
2.6 Rational Functions
Page	Line #	Notes
2	36
2.7 Nonlinear Inequalities
Page	Line #	Notes
2	36