How To Teach with DWFK Precalculus

Chapter 2: Polynomial, Power, and Rational Functions

With the complete vocabulary of functions available to use, the course now turns to a closer focus on particular classes of functions, beginning with polynomials, power functions, and rational functions. Since concepts like asymptotes and end behavior have already been introduced (as well as lines and a first look at parabolas), the presentation has been streamlined, combining two chapters from the previous edition and adding an extended treatment of power functions. There is more algebra in the book from here on, but the graphical and numerical models are continually mentioned in order to reinforce the connections.

Note that the first three sections 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

Objectives
Students will be able to construct and graph linear and quadratic functions and use them to model behavior in real world problems.

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

Teaching 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.

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

Technology Tips
Many students will 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. (Need we say why?) 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 1 in Section 1.6.
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Section 2.2 Power Functions with Modeling

Objectives
Students 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

Teaching Tips
Although half of the "ten 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.

Concavity (discussed in the paragraphs following Example 3) is a geometric property of curves that was not introduced in Section 1.2. Knowing the geometric meaning of concavity will help students in calculus when they analyze it as an application of the second derivative.

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 138, 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

Objectives
Student will be able to graph polynomial functions of degree higher than 2 using graphing technology. They will be able to use the technology to find zeros and relative extrema, analyze end behavior, fit curves to data, and solve modeling problems.

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

Teaching 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. Some students will have used graphers to find extrema in their algebra courses (as such problems now appear regularly in textbooks), but few will 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 189 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 for students trying to study end behavior or search for zeros.

Exercise 75 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

Objectives
Students will be able to divide polynomials using long division or synthetic division. They will be able to analyze the zeros of polynomials using the Factor Theorem, the Rational Zeros Theorem, and the Upper and Lower Bounds tests.

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

Teaching 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 the 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 74.

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.

Be sure to restrict calculator usage 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

Objectives
Students will be able to add, subtract, multiply, and divide complex numbers and write the results in standard form. They will be able to evaluate reciprocals and absolute values of complex numbers and find complex zeros of quadratic functions.

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

Teaching 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 6.5).

This section covers the basic algebra of complex numbers, something that all students should know by the time they get to calculus — despite the fact that they will have little opportunity to use it during their first two calculus courses.

After covering the absolute value of a complex number as thoroughly as you wish, you can bet your students any amount of money that at least one of them, when asked to find on the final exam, will respond with a + bi. You will win.

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.

Before going on to Section 2.6, ask students to explain how the existence of complex zeros limits the calculator's ability to solve equations. It's a good review of many of the important equation-solving concepts covered up to this point. (Some calculators will actually find the complex zeros, which makes for an interesting extension of the conversation.)
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Section 2.6 Complex Zeros and the Fundamental Theorem of Algebras

Objectives
Students will understand the Fundamental Theorem of Algebra and the Linear Factorization Theorem and be able to use them to find real and complex zeros of polynomials with real coefficients.

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

Teaching 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 time to spend covering this section, be advised that the comments in "Teaching 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. Students who solve graphically will miss out on the applications of the theorems.
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Section 2.7 Rational Functions and Equations

Objectives
Students will be able to describe and produce the graphs of rational functions, identify their horizontal and vertical asymptotes, and analyze their end behavior. Students will be able to model real-world problems with rational functions and solve the resulting equations, identifying extraneous solutions when they occur.

Key Ideas
Rational function

Teaching 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. Also, since rational functions arise quite naturally in real-world applications, there are some good modeling problems in the examples and exercises.

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. Students 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 Inequalities in One Variable

Objectives
Students will be able to solve inequalities involving polynomials and rational functions, both algebraically and graphically. They will be able to model real-world problems with such inequalities and solve them.

Key Ideas
Rational inequality
Sign chart

Teaching 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. Students 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

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
Students often forget 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.