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How To Study with DWFK Precalculus

Chapter 2: Polynomial, Power, and Rational Functions

Note that the first three sections of this chapter introduce the function types "with modeling," in keeping with the emphasis on modeling that will be found throughout the course.

Section 2.1	Section 2.4	Section 2.7
Section 2.2	Section 2.5	Section 2.8
Section 2.3	Section 2.6

Section 2.1 Linear and Quadratic Functions with Modeling

Key Ideas

Average rate of change	Linear correlation (positive or negative)
Axis of symmetry of a parabola	Linear depreciation
Coefficient	Linear function
Correlation coefficient	Polynomial function
Degree of a polynomial	Quadratic function
Free-fall	Vertex of a parabola

Study Tips
Emphasizing a linear function as a function with a constant rate of change is an important "pre-calculus" feature of this section. We also exploit the regression capabilities of the calculators to fit linear and quadratic functions to data.

Be careful about assuming that you have seen this material. The underlying algebra (e.g., linear functions, vertex of a parabola) might be familiar to you, but the context in which it is presented will be new to most students. We would expect classes to have the impression that they are covering new ground here.

Technology Tips
You may have calculators that give two forms of the linear regression line: one in the form y = ax + b and one in the form y = a + bx. They might be interested to learn why this redundancy is there. Statisticians are accustomed to the y = a + bx form, so that is what the earlier graphing calculators were programmed to give. When more statistics began to be incorporated into high school algebra, complaints started coming in from algebra teachers that b should be the y-intercept, not the slope. (How dare these calculators mess with slope-intercept form?) Later calculator models were therefore programmed to give an a + bx option for the statisticians and an ax + b option for the algebraists — take your pick!

Note that the quadratic regression gives a value of R² (the coefficient of determination) rather than values of r² and r (the linear correlation coefficient). That is because linear correlation is not appropriate with quadratic regression. See the remarks following Example 6 in Section 1.6.
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Section 2.2 Power Functions with Modeling

You will be able to construct and graph power functions of the form and use them to model behavior in real world problems.

Key Ideas

Concave down	Direct Variation
Concave up	Inverse variation
Constant of proportion	Monomial function
Constant of variation	Power function

Study Tips
Although five of the "twelve basic functions" of Section 1.3 are actually power functions, it took the addition of "power regression" on calculators to earn them the status of their own section in the textbook. Working through this section will leave you wondering why it took so long. Some nice topics like direct and inverse variation and concavity fit comfortably into a unified treatment of this important class of functions.

Example 2 introduces in the manner of the basic functions. It is a good review of the functions concepts to see if students can verify their properties.

Technology Tips
Pay heed to the marginal warning about power regression alongside Example 6. Linear and quadratic regressions can be performed on any set of ordered pairs, but power regressions will fail if all values of x and y are not positive. Several of the other regression models have similar restrictions on x and/or y. (These restrictions are specified in the catalog of regression types on page 149, although they are easy to miss until one actually tries to use the regressions in specific cases.)

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Section 2.3 Polynomial Functions of Higher Degree with Modeling

Key Ideas

Coefficient	Multiplicity of a zero of a polynomial
Cubic function	Polynomial interpolation
Intermediate Value Theorem	Quartic function
Leading term of a polynomial	Term of a polynomial

Study Tips
Although there are formulas for solving cubic and quartic polynomial equations algebraically, they are complicated and must be broken down into different cases. It is not possible to solve a general quintic (5th degree) equation algebraically (a result proved by Niels Henrik Abel at the age of 19), so most modern algebra courses draw the line at the quadratic formula when it comes to exact algebraic solutions. We therefore jump from linear and quadratic polynomials to a section on all the rest, about which there is algebraically less to say until we have acquired the tools of calculus.

The emphasis here is on exploring end behavior, zeros, and relative extrema using grapher technology. You may have used graphers to find extrema in your algebra courses (as such problems now appear regularly in textbooks), but you probably will not have seen the connection between a polynomial's leading term and its end behavior.

Since cubic and quartic regression appear on current calculators, we include some data analysis involving polynomial interpolation. In actual practice, polynomials of higher degree are applied to real-world data less frequently than linear and quadratic models.

Technology Tips
Pay heed to the technology tip on page 195 about how to change horizontal and vertical scales on a calculator when analyzing the graph of a polynomial. "Zooming" in and out will usually change horizontal and vertical scales by equal factors, distorting the shape of the graph—a possible source of frustration when you try to study end behavior or search for zeros.

Exercise 83 in this section is an excellent use of graphing calculators to foreshadow differential calculus. The fact that a smooth curve can be locally approximated by its tangent line at a point underlies most applications of the derivative.
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Section 2.4 Real Zeros of Polynomial Functions

Key Ideas

Factor Theorem	Remainder Theorem
Polynomial division	Synthetic division
Rational Zeros Theorem	Upper and lower bound tests for real zeros

Study Tips
This section contains a few algebraic approaches that have been traditionally used to find zeros of polynomials of degree higher than 2. While they are beautiful results of historical significance, the fact that they can only be used in carefully-constructed cases has caused them to be upstaged in practice by the more generally-applicable technological approach of the previous section.

Students who plan to compete in mathematics competitions will need to know these results well, as they are needed for solving many of the carefully-constructed problems that appear on such examinations. The degree to which this section is emphasized will probably depend on the extent to which your teacher reveres these classical results, but all teachers should keep in mind that there are, realistically, more essential topics yet to come.

Another classical zero-finding theorem, Descartes' Rule of Signs, appears in Exercise 73.

Technology Tips
In is an interesting merger of the modern and the classical, the calculator comes in handy for applying the Rational Zeros Theorem. The most tedious aspect of applying the theorem has always been to "check" the many candidates to see which work. Not only can the calculator be used for those evaluations, but a quick graph can be used to reject most of the candidates before one bothers evaluating them.

The intent of some of the exercises in this section can be compromised if you use a calculator to avoid doing the algebra, so be sure to use your technology only where appropriate. For example, exercises 49-56 should be done algebraically, with the calculator used only as a check.
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Section 2.5 Complex Numbers

Key Ideas

Absolute value (modulus) of a complex number	Imaginary number
Additive identity	Imaginary unit (i)
Additive inverse	Multiplicative identity
Complex conjugate	Multiplicative inverse (reciprocal)
Complex number	Real and imaginary axes
Complex plane	Real and imaginary parts of a number
Discriminant of a quadratic equation	Standard (a + bi) form

Study Tips
The emphasis in a precalculus course is on real-valued functions of real numbers, since those are the functions that can be graphed in the Cartesian plane. Nonetheless, it is natural to discuss complex numbers in a couple of precalculus contexts, one of them being the zeros of polynomial functions. (The other involves trigonometry and will appear in Section 6.6).

This section covers the basic algebra of complex numbers, something that you really ought to know by the time you get to calculus — despite the fact that you will have little opportunity to use it during your first two calculus courses.

Pay close attention to the definition of the absolute value of a complex number, an easy concept for anyone to mess up if you rely on your first instincts. Just as the absolute value of a real number can be thought of as its distance from the origin along a number line, the absolute value of a complex number can be thought of as its distance from (0, 0) in the complex plane. In particular, is not a + bi.

Technology Tips
Modern calculators will do the algebra of complex numbers. If you really want your students to learn how to manipulate complex numbers with pencil and paper (and you probably should), you will want to prohibit the use of calculators on all exercises in this section.
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Section 2.6 Complex Zeros and the Fundamental Theorem of Algebras

Key Ideas

Complex conjugate zeros	Irreducible over the reals
Fundamental Theorem of Algebra	Linear Factorization Theorem

Study Tips
Although the proof of the Fundamental Theorem of Algebra is well beyond the scope of this course, any precalculus student should be able to understand its statement and implications (one implication being the Linear Factorization Theorem). When deciding how much emphasis to place on this section, be advised that the comments in "Study Tips" for Section 2.4 apply equally here.

Technology Tips
"A Word About Proof" following Example 10 in Section 1.1 mentioned the importance of the Fundamental Theorem of Algebra for doing computer (or calculator) searches for zeros of functions. Once we have found n zeros for a polynomial of degree n, this is the theorem that tells us to stop searching. Similarly, until we have found n zeros (including complex and/or repeated zeros), this is the theorem that tells us to keep searching.

This is another section in which the manipulative exercises should be solved algebraically, then verified graphically. If you solve these problems graphically, you will miss out on the applications of the theorems.
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Section 2.7 Graphs of Rational Functions

Key Ideas
Rational function

Study Tips
This section is a good review of many of the function concepts of Chapter 1 and should be approached in that spirit. Asymptotes, zeros, and intercepts can be found algebraically and then used to give a full geometric picture (graph) of the function's behavior.

Technology Tips
Although calculators are not specifically prohibited in most of the Section 2.7 Exercises, the clear intent of Exercises 1-48 is that they not be done with graphing calculators. You should be able to make the connection between the geometric behavior and the algebraic expressions without actually seeing the graphs.
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Section 2.8 Solving Equations in One Variable

Objectives
You will be able to solve rational equations both algebraically and graphically. You will be able to eliminate extraneous solutions. You will be able to model real-world problems with rational functions and solve the resulting equations.

Key Ideas
Rational equations
Extraneous solutions

Study Tips
The concept in this section is one found in most first year Algebra textbooks. There is a fair amount of work in solving rational equations. Many times, you will be left with a quadratic equation that needs to be solved. Skill in this regard is necessary.
Since rational functions arise quite naturally in real-world applications, there are some good modeling problems in the examples and exercises.

Technology Tips
Since the work in solving rational equations can be somewhat laborious; it is useful to confirm your solutions graphically. Comparison of the algebraic and graphical solutions will show the existence of extraneous solutions. Technology should be used to model the real-world problems.

Section 2.9 Solving Inequalities in One Variable

Key Ideas
Rational inequality
Sign chart

Study Tips
This section is more important than it looks, as creating sign graphs for functions is a skill required in several contexts in a calculus course. You might be used to "plugging in" numbers to determine the sign of a function on an interval, but we feel that the method illustrated in this section is better for several reasons. First, it is a waste of time to compute an actual value when all that is needed is its sign, and second, analyzing sign changes at zeros reinforces the useful notion that a polynomial graph "behaves" near a zero like a monomial graph behaves near x = 0. For example, the graph of x = 0.

behaves near x = 1 like the parabola behaves near 0 (so there is no sign change there), while it behaves near x = 2 like the cubic behaves close to 0 (so there is a sign change there).

Technology Tips
Always keep in mind how simple it can be to solve inequalities algebraically. An inequality like

used to be worthy of being a bonus problem on a mathematics competition. Such inequalities can be solved today at a glance with a graphing calculator.