MATH STUDENT BOOK. 12th Grade Unit 2

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1 MATH STUDENT BOOK 12th Grade Unit 2

2 Unit 2 FUNCTIONS MATH 1202 FUNCTIONS INTRODUCTION 3 1. LINEAR FUNCTIONS 5 GRAPHS OF LINEAR FUNCTIONS 5 EQUATIONS OF LINEAR FUNCTIONS 10 SELF TEST 1: LINEAR FUNCTIONS SECOND-DEGREE FUNCTIONS 17 SOLUTIONS TO SECOND-DEGREE FUNCTIONS 17 RELATIONSHIPS BETWEEN ZERO AND COEFFICIENTS 22 QUADRATIC INEQUALITIES 31 SELF TEST 2: SECOND-DEGREE FUNCTIONS POLYNOMIAL FUNCTIONS 37 INTRODUCTION TO POLYNOMIAL FUNCTIONS 37 Nth-DEGREE EQUATIONS 2 SOLVING POLYNOMIAL EQUATIONS 6 SELF TEST 3: POLYNOMIAL FUNCTIONS 50. COMPLEX NUMBERS 53 INTRODUCTION TO COMPLEX NUMBERS 53 OPERATIONS USING COMPLEX NUMBERS 58 CONJUGATES AND POLYNOMIAL IDENTITIES 63 DISTANCE AND MIDPOINT 66 SELF TEST : COMPLEX NUMBERS SPECIAL FUNCTIONS 71 RATIONAL INEQUALITIES 71 GREATEST INTEGER FUNCTION 76 EXPONENTIAL FUNCTIONS 80 LOGARITHMIC FUNCTIONS 85 FUNCTION COMBINATIONS 90 SELF TEST 5: SPECIAL FUNCTIONS REVIEW FUNCTIONS 99 GLOSSARY 102 1

3 FUNCTIONS Unit 2 LIFEPAC Test is located in the center of the booklet. Please remove before starting the unit. Author: Alpha Omega Publications Editors: Alan Christopherson, M.S. Lauren McHale, B.A. Media Credits: Page 30: Comstock, Stockbte, Thinkstock; 39: Danler, istock, Thinkstock. 80 N. 2nd Ave. E. Rock Rapids, IA MMXVII b Alpha Omega Publications, a division of Glnlon, Inc. All rights reserved. LIFEPAC is a registered trademark of Alpha Omega Publications, a division of Glnlon, Inc. All trademarks and/or service marks referenced in this material are the propert of their respective owners. Alpha Omega Publications, a division of Glnlon, Inc., makes no claim of ownership to an trademarks and/or service marks other than their own and their affiliates, and makes no claim of affiliation to an companies whose trademarks ma be listed in this material, other than their own. 2

4 Unit 2 FUNCTIONS Functions Introduction This LIFEPAC reviews various tpes of functions, including linear, quadratic, polnomial, eponential, and logarithmic functions. You will learn how to graph each of these tpes of functions and solve equations and inequalities in which the appear. You will also learn about comple numbers and their importance in solving polnomial equations. Objectives Read these objectives. The objectives tell ou what ou will be able to do when ou have successfull completed this LIFEPAC. When ou have finished this LIFEPAC, ou should be able to: 1. Identif and solve polnomial, eponential, and logarithmic equations. 2. Graph polnomial, eponential, and logarithmic equations. 3. Recognize various transformations to basic polnomial, eponential, and logarithmic functions and draw appropriate graphs.. Use snthetic division to find the roots of a polnomial equation, as well as upper and lower limits on those roots. 5. Geometricall represent basic arithmetic operations of comple numbers on the comple plane. 6. Convert between rectangular and polar forms of comple numbers. 7. Use the comple conjugate to simplif fractions and find the modulus of a comple number. 8. Use interval notation to epress the solution to polnomial and rational inequalities in mathematical and real-world contets. 9. Identif and graph the greatest integer function. 10. Combine functions using basic arithmetic operations. Introduction 3

5 FUNCTIONS Unit 2 Surve the LIFEPAC. Ask ourself some questions about this stud and write our questions here. Introduction

6 Unit 2 FUNCTIONS 1. LINEAR FUNCTIONS GRAPHS OF LINEAR FUNCTIONS As we continue studing the concept of functions, we will first look at the polnomial functions of degrees one and two. After that, we will stud higher-degree polnomial functions. Finall, we will stud several special functions including the greatest integer function, the eponential function, the logarithmic function, and combinations of these three functions. Section Objectives Review these objectives. When ou have completed this section, ou should be able to: Identif polnomial functions. Solve linear polnomial functions. Vocabular Stud these words to enhance our learning success in this section. linear function A polnomial function of the form = a 0 n + a 1 n 1 + a 2 n a n 1 + a n, where a 0 0, and n = 1. -intercept The value of when = 0; the value of the first coordinate of the point where the line crosses the -ais. -intercept The value of when = 0; the value of the second coordinate of the point where the line crosses the -ais. zero of a function A value of for which the function value is zero; a root or solution of f() = 0. Note: All vocabular words in this LIFEPAC appear in boldface print the first time the are used. If ou are not sure of the meaning when ou are reading, stud the definitions given. WHAT ARE POLYNOMIAL FUNCTIONS? The polnomial functions are a group of functions in the form = a 0 n + a 1 n 1 + a 2 n a n 1 + a n, where a 0, a 1, a 2,... are numerical coefficients and n, a positive integer, is the eponent of the variable. The number n then becomes the degree of the polnomial. For eample, a second-degree polnomial would have the form a a 1 + a 2, where a 0, a 1, and a 2 are real-number coefficients. If a 0 = 2, a 1 =, and a 2 = -3, the polnomial would be: SOLVING LINEAR POLYNOMIAL FUNCTIONS If a 0 0, and n = 1, the epression is a linear equation or linear function. Its degree is one, so ou can also call it a first-degree polnomial. A linear function in is one that can be written in the standard form = m + b, m 0, and where m and b are constants. Section 1 5

7 FUNCTIONS Unit 2 Some eamples of linear functions are = 3 + 2, + = 6, and 7 = 3. For practice, identif a 0, a 1, and n in each of these eamples: 1. = = = 3 You should have found the following: 1. a 0 = 3, a 1 = 2, n = 1 2. a 0 = -1, a 1 = 6, n = 1 3. a 0 = 7 3, a 1 = 0, n = 1 It is important to remember that the graph of a linear function is a straight line, but not all lines are linear functions. For eample, a vertical line is not a linear function because of how functions are defined: if two or more points of a graph lie on the same vertical line, then the does not determine a unique value and the relation is not a function. Also, a linear function is a first-degree polnomial. Eample Graph + = 2. Find at least three solutions. + = 2. Solve for. = Here are three possible solutions: Locate the solutions as points on the coordinate plane. Do the match the following sample? When working with functions, there will be certain solutions that will be important for ou to identif. With linear equations, it is important that ou identif the zeros of the function. The zero of a function is a value of for which the function f() = 0. Thus, the zeros of a function are roots or solutions of f() = 0. The zero of the function in the preceding eample, + = 2, or, the value of when = 0, is 2. That is, f() = 0 when = 2. This value, (2, 0), is also called the -intercept and is the point where the line crosses the -ais. Similarl, we can solve for the point where the line crosses the -ais. This value, (0, 2), is called the -intercept. -7 Root, -intercept, and zero of the function all refer to the same point. You can tell it s important, since we have so man names for it! Usuall, we sa intercept when we are talking about a graph, and root or zero when we are finding the value using algebra. If ou are asked for the intercept, ou should provide both coordinates. In our eample, the -intercept is (0, 2). SOLVING LINEAR EQUATIONS THAT INCLUDE FRACTIONS (0, 2) A special case of a linear function that we need to consider is in the form of a fraction. Eample -intercept (2, 0) -intercept Graph = 2 9. Find at least three solutions Implied in this rule is that -3; therefore, the domain of this function is the set of all real numbers ecept = -3. If we eclude = -3, then we ma simplif = 2 9 in the following + 3 manner ( 3)( + 3) Factoring the numerator, we get =, ( + 3) which can be reduced to = 3. É 6 Section 1

8 Unit 2 FUNCTIONS Since = 3 is a straight line and -3, we have a line with a hole in it. Three solutions are: The zero of this function is 3; that is, f() = 0 when = 3. The value of the function at = -3 is undefined. When ou take calculus, ou will be able to prove that the closer the value of gets to -3, the closer the function value,, will get to The graph of this function looks like this: Multiple-choice questions are presented throughout this unit. To enhance the learning process, students are encouraged to show their work for these problems on a separate sheet of paper. In the case of an incorrect answer, students can compare their work to the answer ke to identif the source of error. Complete the following activities. 1.1 _ A function of the form = a 0 n + a 1 n 1 + a 2 n a n 1 + a n, where a 0 0, and n = 1, is called a _. a. second-degree polnomial b. root c. linear equation d. -intercept 1.2 _ Which of the following is a third-degree polnomial? _ a. = 2 5 b. = c. = d. = _ Find three solutions for the following linear function and use them to draw its graph. _ F() = + 3 Section 1 7

9 FUNCTIONS Unit 2 1. _ Find three solutions for the following linear function and use them to draw its graph. _ F() = _ Find three solutions for the following linear function and use them to draw its graph. _ F -1 () if F() = _ If F() = + 3, what is the zero of F -1 ()? _ 1.7 _ What is the -intercept of G() if G() = 3 2? _ 1.8 _ Find three solutions for the following linear function and use them to draw its graph. _ g() = Section 1

10 Unit 2 FUNCTIONS 1.9 _ Find three solutions for the following linear function and use them to draw its graph. h() = _ Find three solutions for the following linear function and use them to draw its graph. A() = If G() = 3 5, then G -1 () = _. a. 5 3 b c d What is the zero of the function A() = ? 1.13_ What is the -intercept of h() = 2 16? _ Section 1 9

11 MAT1202 Jul 18 Printing 80 N. 2nd Ave. E. Rock Rapids, IA ISBN

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