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175

If you ride the Giant Dipper roller coaster, you’ll climb to high points and descend to low points along a continuous track. If the track does not loop or overlap, then its shape can be modeled by a function. In this chapter you will study polynomial functions, and learn how the degree and the coe� cients of polynomial functions determine the shape of their graphs. � e graphs of polynomial functions o� en have hills and valleys, and sometimes resemble the track of a roller coaster.

Polynomial and Rational FunctionsPolynomial and Rational Functions

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CHAP TE R O B J EC TIV ES

• Discover some properties of polynomial and rational functions and their graphs.

• Review the properties, graphs, and techniques associated with quadratic functions.

• Given a polynomial function, determine from the graph what degree it might be, and vice versa.

• Given a polynomial function, fi nd the zeros from the equation or the graph.

• Given a set of points, fi nd the equation for the polynomial function that fi ts the data exactly or fi ts best for a given degree.

• Discover some properties of polynomial and rational functions and their graphs.

• Multiply, divide, add, and subtract rational expressions, and resolve proper algebraic fractions into the sum of two or more partial fractions.

• Given a rational algebraic function f, fi nd x for a given value of f (x).

• 175A Chapter 4 Interleaf: Polynomial and Rational Functions

Overview In this chapter students signifi cantly expand what they have learned about polynomial and rational functions. Factoring, completing the square, and the quadratic formula are reviewed. An examination of all zeros of the quadratic function leads to an investigation of real and nonreal complex zeros, the fundamental theorem of algebra, and graphs of higher-degree polynomial functions. Students study rational functions, asymptotes, and discontinuities. Finally, partial fractions and fractional equations complete the chapter.

A cumulative review of Chapters 1–4 appears in Section 4-9.

Using This Chapter Th is chapter focuses on polynomial and rational functions, the corresponding polynomial and rational expressions, and polynomial and rational equations. Th e study of functions includes an examination of the graphs, asymptotes, discontinuities, and fi tting of functions to data. Th e study of expressions involves factoring, completing the square, operations with rational expressions, and partial fractions. Th e study of equations includes the quadratic formula, complex numbers, fractional equations, and extraneous solutions. Th e relationships between the zeros and the graph of a function are emphasized throughout the chapter. Portions of this chapter review topics students learned in second-year algebra. You may prefer to select only a few problems from the familiar material and concentrate on the new content in the chapter.

Teaching Resources Explorations Exploration 4-3: Graphs and Zeros of Cubic Functions Exploration 4-3a: Cubic Function Graphs Exploration 4-3b: Synthetic Substitution Exploration 4-3c: Sum and Product of the Zeros of a Polynomial Exploration 4-4a: Fitting a Polynomial Function to Points Exploration 4-5: Transformations of the Parent Reciprocal

Function Exploration 4-5a: Rational Functions and Discontinuities Exploration 4-6a: Introduction to Partial Fractions Exploration 4-7a: Fractional Equations

Blackline Masters Sections 4-8 and 4-9

Assessment Resources Test 8, Sections 4-1 to 4-4, Forms A and B Test 9, Sections 4-5 to 4-7, Forms A and B Test 10, Chapter 4, Forms A and B Test 11, Cumulative Test, Chapters 1–4, Forms A and B

Technology Resources Sketchpad Presentation Sketches Polynomial Fit Present.gsp

Activities CAS Activity 4-2a: Coeffi cients of Quadratic Functions CAS Activity 4-5a: End Behavior of a Rational Function

Calculator Programs SYNSUB

Polynomial and Rational FunctionsPolynomial and Rational FunctionsC h a p t e r 4

• Chapter 4 Interleaf 175B

Standard Schedule Pacing Guide

Block Schedule Pacing Guide

Day Section Suggested Assignment

1 4-1 Introduction to Polynomial and Rational Functions 1–8

2 4-2 Quadratic Functions, Factoring, and Complex Numbers

RA, Q1–Q10, 1, 5, 7, 9, 13–19 odd, 23, 25, 28 3 33–37 odd, 41, 43, 47, 53, 55, 56, 57, 61, 63, 65b, 65d, 66 4

4-3 Graphs and Zeros of Higher-Degree Functions RA, Q1–Q10, 5–9, 11–19 odd

5 21, 25, 29, 30, 35–38

6 4-4 Fitting Polynomial Functions to Data RA, Q1–Q10, 1, (3), 4, 5, 7–9, 13, 15, (16)

7 4-5 Rational Functions: Asymptotes and Discontinuities

RA, Q1–Q10, 1–5, 7, 9, 13–21 odd 8 23–31 odd, 35, 36

9 4-6 Partial Fractions and Operations with Rational Expressions RA, Q1–Q10, 7, 9, 13, 16, 21, 25, 29, 31, 41

10 4-7 Fractional Equations and Extraneous Solutions RA, Q1–Q10, 1–3, 5–11 odd, 15–27 odd

11 4-8 Chapter Review and Test

R0–R7, T1–T19 12 Section 4-9 Problems 1–13 13

4-9 Cumulative Review, Chapters 1–4 14–29

14 30–37

Chapter 4 Interleaf 175B

Day Section Suggested Assignment

1 4-2 Quadratic Functions, Factoring, and Complex Numbers RA, Q1–Q10, 1, 5, 7, 9, 13–19 odd, 23, 25, 28, 33–37 odd, 41, 43, 47, 53

2 4-3 Graphs and Zeros of Higher-Degree Functions RA, Q1–Q10, 5–9, 11–21 odd, 25, 29, 30, 35–38

3 4-4 Fitting Polynomial Functions to Data RA, Q1–Q10, 1, 4, 5, 7–9, 13, 15

4-5 Rational Functions: Asymptotes and Discontinuities RA, Q1–Q10, 1–5, 7, 9, 13–21 odd

4 4-5 Rational Functions: Asymptotes and Discontinuities 23–31 odd, 35, 36

5 4-6 Partial Fractions and Operations with Rational Expressions RA, Q1–Q10, 7, 9, 13, 16, 21, 25, 29, 31, 41

6 4-7 Fractional Equations and Extraneous Solutions RA, Q1–Q10, 1–3, 5–11 odd

7 4-7 Fractional Equations and Extraneous Solutions 15–27 odd

4-8 Chapter Review and Test R0–R7, T1–T19

8 4-8 Chapter Review and Test Section 4-9 Problems 1–13

• 177Section 4-1: Introduction to Polynomial and Rational Functions

Introduction to Polynomial and Rational Functions Recall from Section 1-2 that a polynomial function has an equation such as f (x) x 2 2x 8 where no operations other than addition, subtraction, or multiplication are performed on the independent variable x. (Squaring is equivalent to multiplying.) Function f is a quadratic function because the highest power of x is the second power. A rational function has an equation such as g (x) x

3 5x 2 2x 25 _____________ x 3 , where the y-value is a ratio of two polynomials—in this case a cubic polynomial divided by a linear polynomial. Figures 4-1a and 4-1b show the graphs of functions f and g.

x 422

10 f(x)

10

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10 g(x)

10 Asymptote

Figure 4-1a Figure 4-1b

Discover some properties of polynomial and rational functions and their graphs.

Introduction to Polynomial and Rational Functions

4 -1

Discover some properties of polynomial and rational functions and their graphs.

Objective

1. Function f appears to have x-intercepts at 2 and 4. Con� rm by direct substitution that f ( 2) and f (4) both equal zero. � en explain why 2 and 4 are also called zeros of function f.

2. � e x-value 2 appears to be a zero of function g. Con� rm or refute this observation numerically.

3. Function g has a vertical asymptote at x 3. Try to � nd g (3). � en explain why g (3) is unde� ned. Explain the behavior of g (x) when x is close to 3 by � nding g (3.001) and g (2.999).

4. Plot the graphs of functions f and g on the same screen. What do you notice about the two graphs when x is far from 3? When x is close to 3? How could you describe the relationship of the graph of function f to the graph of function g?

5. Let function h be the vertical translation of function f by 10 units. � at is, h(x) f (x) 10. Show by graphing that no zeros of function h are real numbers.

6. Explain why a quadratic function can have no more than two real numbers as zeros.

7. As a challenge, show that the complex number 1 i (where i

___ 1 ) is a zero of h (x).

Explain why a function can have a zero that is not an x-intercept.

8. What did you learn as a result of doing this problem set?

1. Function f appears to have f appears to have f x-intercepts at 2 5. Let function h be the vertical translation of

Exploratory Problem Set 4–1

Chapter 4: Polynomial and Rational Functions176

In this chapter you’ll learn about polynomial functions. You’ll � nd the zeros of these functions, a generalization of the concept of x-intercept. � e techniques you learn will allow you to analyze rational functions, in which f (x) is a ratio of two polynomials. � ese ratios can represent average rates of change. You’ll investigate these concepts in four ways.

� e graph of the polynomial function f (x) x 4 8 x 2 8x 15 has two x-intercepts.

f (x) x 4 0 x 3 8 x 2 8x 15 (x 1)(x 3)(x 2 i)(x 2 i)

f (1) 0, f (3) 0, f ( 2 i) 0, and f ( 2 i) 0

Zeros of the function are x 1, x 3, x 2 i, and x 2 i.

A fourth-degree function has exactly four zeros, counting multiple zeros, i f the domain of the func