Intermediate Algebra BARBARA GOLDNER EDGAR JASSO DEANNA LI PAM LIPPERT. North Seattle College

Transcription

1 Intermediate Algebra BARBARA GOLDNER EDGAR JASSO DEANNA LI PAM LIPPERT North Seattle College Second Edition Fall 2016

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3 TABLE OF CONTENTS About this Book INTRODUCTION TO FUNCTIONS What is a Function Multiple Representations of functions Function Notation Domain and Range Applications of Functions Practice Problems Chapter 1 Assessment FUNCTIONS AND FUNCTION OPERATIONS Combining Functions Applications of Function Operations Practice Problems Chapter 2 Assessment LINEAR EQUATIONS AND FUNCTIONS Linear Equations and Functions Graphs of Linear Equations Horizontal and Vertical Lines Writing the Equation of a Line Systems of Linear Equations Practice Problems Chapter 3 Assessment LINEAR FUNCTIONS AND APPLICATIONS Review of Linear Functions Average Rate of Change Determining the Equation of a Line Scatterplots and Linear Modeling Practice Problems Chapter 4 Assessment ABSOLUTE VALUE EQUATIONS AND INEQUALITIES Absolute Value Equations Absolute Value Inequalities Absolute Value Applications

4 4 TABLE OF CONTENTS Practice Problems Chapter 5 Assessment POLYNOMIAL EQUATIONS Factoring Expressions Solving Equations by Factoring Completing the Square The Quadratic Formula Complex Numbers Complex Solutions to Quadratic Equations Applications of Quadratic Equations Practice Problems Chapter 6 Assessment GRAPHS OF QUADRATIC FUNCTIONS Solutions to Quadratic Equations Characteristics of Quadratic Equations Solving Quadratic Equations Algebraically & Graphically Applications of Quadratic Functions Quadratic Modeling Practice Problems Chapter 7 Assessment INTRODUCTION TO EXPONENTIAL AND LOGARITHMIC FUNCTIONS Linear functions vs Exponential Functions Introduction to Logarithms Computing Logarithms Solving Logarithmic Equations Applications of Logarithmic Equations Solving Exponential Equations Applications of Exponential Functions Practice Problems Chapter 8 Assessment RADICAL FUNCTIONS Roots, Radicals & Radical Expressions Operations with Radical Expressions Solving Radical Equations Practice Problems Chapter 9 Assessment RATIONAL FUNCTIONS Operations with Rational Expressions Solving Rational Equations

5 TABLE OF CONTENTS Applications of Rational Expressions Practice Problems Chapter 10 Assessment COURSE REVIEW Overview of Functions Solving Equations Applications Practice Problems

6 6 TABLE OF CONTENTS

7 ABOUT THIS BOOK 7 ABOUT THIS BOOK Mathematics instructors from North Seattle College (NSC) created this workbook to better serve our students and to align our curriculum with NSC learning outcomes. We want our students at this level to experience algebra through functions, their graphs and with interesting applications. We also hope our students will develop a deeper understanding of algebraic concepts without the aid of graphing calculators. This workbook is organized as follows: Chapters Lessons are the main instructional component for each chapter. They are where ideas are introduced. Examples provide further explanation of a concept. It is recommended that students read through these examples carefully. Problems can be done by watching online videos or used as classroom examples by instructors. Exercises are for student practice to help reinforce concepts. It is recommended to solve all these problems and to do so in the order they appear, showing as much work as possible in a neat and organized fashion. There is space provided to work the solutions in the workbook. Practice Problems There is a Practice Problems section at the end of each chapter. The only way to learn math is by practice. To be successful in this course, students should work all the practice problems. We suggest that students attempt them on their own first before seeking help. Although this is a workbook, and there is space to show most of the work, we recommended that students keep an organized notebook for this class. Students should do the practice problems in this notebook, showing all the worked solutions in a neat and organized way, so they can refer to them easily. Solutions to all the Practice Problems have been compiled and each instructor can make them available to the students. Chapter Assessments The last part of each Chapter is a short assessment. Some instructors may requiere these assessments to be completed as part of the course. Even if that is not the case, these assessments can serve studnets so that they can recognize which topics of a chapter may need further review. WAMAP (online Homework Assessment System) Some instructors may use WAMAP as an online homework platform in conjunction with this workbook.

8 8 TABLE OF CONTENTS The Math faculty at NSC would like to thank the faculty at Scottsdale Community College (SCC) for the foundation of this workbook. The NSC Math faculty would also like to thank the Math 098 students of Winter 2015 for their patience as the book was being written. Intermediate Algebra is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. ( Based on a work by Scottsdale Community College.

9 CHAPTER 1 INTRODUCTION TO FUNCTIONS Throughout this class, we will be looking at various algebraic functions and the characteristics of each. Before we begin, we need to learn what a function is and look at the rules that a function must follow. We also need to investigate the different ways that we can represent a function. It is important that we go beyond simple manipulation and evaluation of these functions by examining their characteristics and analyzing their behavior. Looking at the function modeled as graphs, tables, and sets of ordered pairs is critical to accomplishing this goal. Chapter Topics: Section What is a Function? Definition of Function Independent and Dependent Variables Section Multiple Representation of Functions Sets of Ordered Pairs (Input, Output) Tables Graphs Vertical Line Test Behavior of Graphs Section Function Notation Function Evaluation Working with Input and Output Multiple Representations Graphs and Tables Section Domain and Range Definitions Multiple Representations Restricting Domain and Range Section Applications of Functions Criteria for a Good Graph Practical Domain and Range 9

10 10 INTRODUCTION TO FUNCTIONS 1.1. WHAT IS A FUNCTION What is a Function? The concept of a function is one that is very important in mathematics. The use of this term is very specific and describes a particular relationship between two quantities: an input quantity and an output quantity. Specifically, a relationahip between two quantities can be defined as a function if each input value is associated with one and only one output value. Why do we care about Functions? Imagine that you are a nurse working the emergency room of a hospital. A very sick person arrives. You know just the medicine needed but you are unsure of the exact dose. First, you determine the patient's weight (200 pounds). Then you look at the table and see the given dosage information: Weight (in lbs.) Dosage (in ml) You are immediately confused and very concerned. How much medicine should you give? 10 ml or 100 ml? One amount could be too much and the other may not be enough. How do you choose the correct amount? What you have here is a situation that does not define a function. In this case, for the input value 200 lbs, there are two choices for the output value. If you have a function, you will not have to choose between output values for a given input. In the real case of patients and medicine, dosage charts are based upon functions. A More Formal Definition of Function: A function is a rule between numbers that assigns a unique output value to each input value.

11 1.1 WHAT IS A FUNCTION 11 Problem 1 : Class/Media - Do the Data Represent a Function The table below gives the height, H, in feet, of a golf ball t seconds after being hit. t = Time H = Height (in seconds) (in feet) a) Identify the input quantity (include units). b) Identify the input variable. c) Identify the output quantity (inlcude units). d) Identify the output variable. e) Write the data as a set of ordered pairs. f) Interpret the meaning of the ordered pair (3, 144). g) Is the height of the golf ball a function of time? Why or why not? h) Is time a function of the height of the golf ball? Why or why not?

12 12 INTRODUCTION TO FUNCTIONS Example 1 Investigating Functional Relationship Let us investigate the functional relationship between the two quantities, Numerical Grade and Letter Grade. First, let Numerical Grade be the input quantity and Letter Grade be the output quantity. Below is a sample data set that is representative of the situation. Numerical Grade Letter Grade 95 A 92 A 85 B 73 C The numbers above are made up to work with this situation. Other numbers could have been used. We are assuming a standard 90, 80, 70, etc. grading scale. Hopefully, you can see from this data that no matter what numerical value we have for input, there is only one resulting letter grade. Notice that the repeated outputs A are not a problem since the inputs are different. You can uniquely predict the output for any numerical grade input. So, from this information, we can say that Letter Grade (output) is a function of Numerical Grade (input). Now let us switch the data set above. Letter Grade Numerical Grade A 95 A 92 B 85 C 73 Can you see there is a problem here? If you say that you have an A in a class, can you predict your numerical grade uniquely? No. There are a whole host of numerical scores that could come from having an A. The same is true for all the other letter grades as well. Therefore, Numerical Grade (output) is not a function of Letter Grade (input). Summary: Letter Grade is a function of Numerical Grade Numerical Grade is not a function of Letter Grade

13 1.1 WHAT IS A FUNCTION 13 Terminology In the language of functions, we call the independent variable the variable that is used as an input and the dependent variable the one thata is the output. The dependent variable (or output) is a function of or depends on the independent variable (or input). Exercise 1 Do the Data Represent a Function? The table below gives the value of a car n years after purchase. n = Time (in years) V= Value (in dollars) 0 $32,540 1 $28,310 2 $24,630 3 $21,428 4 $18,642 5 $16,219 6 $14,110 a) Identify the input quantity (include units). b) Identify the output quantity (include units). c) Identify the dependent variable. d) Identify the independent variable. e) Interpret the meaning of the ordered pair (2, $24,630). f) Is the value of the car a function of time? Why or why not?

14 14 INTRODUCTION TO FUNCTIONS 1.2. MULTIPLE REPRESENTATIONS OF FUNCTIONS Problem 1 : Class/Media - Determine Functional Relationship Using Multiple Representation Which of the following represent a functional relationship? SETS OF ORDERED PAIRS (Input, Output) a) {(3,2), (5,0), (4, 7)} b) {(0, 2), (5, 1), (5, 4)} c) {( 3,2), (5,2), (4,2)} TABLES d) e) f) x y x y x y GRAPHS g) h)

15 1.2 MULTIPLE REPRESENTATIONS OF FUNCTIONS 15 THE VERTICAL LINE TEST If all vertical lines intersect the graph of a relation at only one point, the relation is a function. One and only one output value exists for each input value. If any vertical line intersect the graphs of a relation at more than one point, the relation fails the vertical line test and is not a function. More than one output value exists for some (or all) input value(s). Example 1 Determine Functional Relationships Using Multiple Representations Set of Ordered Pairs Function {( 7, 6),( 3, 3),(1, 8),(5, 8),(11, 0)} No input value is repeated in an ordered pair NOT a Function {(8,3), (6,1), (8, 1), (6,11), (2, 5)} Two of the listed input values (6 & 8) are paired with more than one output value Table x y All input values are associated with one unique output x y Two of the listed input values (1 and 4) are associated with more than one output value Graph No vertical line intersects the graph in more than one point. We say the graph passes the vertical line test. There is at least one vertical line that intersects the graph at more than one point. This corresponds to some inputs having more than one output value. The graphs fails to pass the vertical line test.

16 16 INTRODUCTION TO FUNCTIONS Exercise 1 Representaitons Which of the following represent a functional relationship? a) b) c) {(4, 1),( 7, 1),( 3, 1),(0, 1)} x y d) e) f) {(3,5), ( 3,6), (8,5), (2, 6)} x y Problem 2 : Class/Media - Does the Statement Describe a Function? Explain your choice for each of the following. Remember when the word function is used, it is in a purely mathematical sense, not in an everyday sense. a) Is the number of children a person has a function of their income? b) Is your weekly pay a function of the number of hours you work each week? (Assume you work an hourly rate job with no tips)

17 1.2 MULTIPLE REPRESENTATIONS OF FUNCTIONS 17 A function is: Increasing if the output gets larger, as the input gets larger. Decreasing if the output gets smaller, as the input gets larger. Constant if the output does not change, as the input changes. Note. When you read graphs, read them like a book... from left to right Example 2 Behavior of Functions a) The following functions are increasing. x y b) The following functions are decreasing. x y c) The following functions are constant. x y

18 18 INTRODUCTION TO FUNCTIONS 1.3. FUNCTION NOTATION FUNCTION NOTATION is used to indicate a functional relationship between two quantities as follows: Function Name (input value) = output value The statement f (x) = y refers to the function, f, and corresponds to the ordered pair (x, y), where x is the input variable and y is the output variable. Function Evalutation: To evaluate a function at a particular value of the input variable, replace each occurrence of the input variable with the given value and compute the result. Note. Use of parenthesis around your input value, especially if the input is a negative value, can help achieve correct results. Problem 1 : Class/Media - Function Evaluation Given f (x) = 2x 5, evaluate the following: a) f (2) b) f ( 1) c) f (x + 1) d) f ( x)

19 1.3 FUNCTION NOTATION 19 Example 1 Functional Evaluation Given f (x)= 5x 2 3x 10, evaluate the following: a) f (2) b) f ( a) Solution. a) f (2) = 5(2) 2 3(2) 10 = 5(4) 3(2) 10 = = = 4 b) f ( a) = 5( a) 2 3( a) 10 = 5a 2 3( a) 10 = 5a 2 + 3a 10 Note. Follow the order of operation in evaluating functions. When working with functions, there are two main questions we will ask and solve as follows: Given a particular input value, what is the corresponing output value? Given a particular output value, what is the corresponding input value? Problem 2 : Class/Media - Working with Input and Output Given f (x)= 2x + 5, determine each of the following. Write your answers as ordered pairs. Given the input, find the output. a) Find f (0) b) Find f ( 2) Given the output, find the input. c) Find x if f (x)= 7 d) Find x if f (x) = 11

20 20 INTRODUCTION TO FUNCTIONS Exercise 1 Working with Input and Output Given f (x) = 3x 4, compute each of the following. Show all steps and write your answers as ordered pairs. Write answers as integers or reduced fractions (no decimals). a) Find f (2) b) Find x if f (x)= 7 c) Find f ( 3) d) Find x if f (x)= 12 e) Find f ( x) f) Find f (x 5) Problem 3 : Class/Media - Working with Function Notation Using a Set of Ordered Pairs The function g(x) is shown: g= {(1,3), (5,2), (8,3), (6, 5)} a) Find g(1). b) Find x if g(x)= 5. c) Find x if g(x)= 3.

21 1.3 FUNCTION NOTATION 21 Problem 4 : Class/Media - Working with Function Notation Using a Table The function V(n) gives the value, V, of an investment (in thousands of dollars) after n months. n V(n) a) Identify the input quantity (include units). b) Identify the output quantity (include units). c) Write a sentence explaining the meaning of the statement V(1)= d) Determine V(3) and write a sentence explaining its meaning. e) For what value of n is V(n)= 3.02? Interpret your answer in a complete sentence.

22 22 INTRODUCTION TO FUNCTIONS Problem 5 : Class/Media - Working with Function Notation Using a Graph The graph of the function D(t) below shows a person's distance from home as a function of time. a) Identify the input quantity (include units). b) Identify the output quantity (include units). c) Write a sentence explaining the meaning of the statement D(15)= 10. d) Determine D(0) and write a sentence explaining its meaning. e) For what value of t is D(t)= 0? Interpret your answer in a complete sentence.

23 1.3 FUNCTION NOTATION 23 Exercise 2 Create a Table and Graph of a Function Consider the function f (x) = 5 2x. a) Complete the table. x f (x) b) Sketch the graph of f (x)= 5 2x.

24 24 INTRODUCTION TO FUNCTIONS 1.4. DOMAIN AND RANGE The set of all possible input values for the independent variable of a function is called the domain. The domain can be listed or expressed in inequality notation or interval notation. The values generated by substituting the domain into the function are the range of the function. Suppose that instead of purchasing a bus pass, you decide to pay per ride. Each ride costs $2.50. The domain in this case will not include negative numbers because it does not make sense to ride negative rides. In addition, whether you ride the bus for one block or all the way to your destination, it is considered one full ride. Therefore, the values of the independent variable will be the set of whole numbers. These input values are referred to as the practical domain for this particular situation. Domain: All whole numbers or {0, 1, 2, 3,.} The cost of the ride depends upon the number of rides times $2.50 or the set of whole numbers times $2.50. Therefore, the range in this situation are positive multiples of $2.50. Range: All positive multiples of $2.50 or {0, 2.50, 5.00, 7.50, } The domain of a function is the set of all possible values for the input quantity. The range of a function is set of all possible values for the output quantity. Problem 1 : Class/Media - Domain and Range, Multiple Representations SET OF ORDERED PAIRS a) Determine the domain and range of the function P(x) = {(2, 3), (4, 5), (6, 0), (8, 5)}. Domain: Range:

25 1.4 DOMAIN AND RANGE 25 TABLE b) Determine the domain and range of the function R(t) defined below. t R(t) Domain: Range: GRAPH c) Determine the domain and range of the function g(x) defined below. Domain of g(x): Inequality Notation: Interval Notation: Range of g(x): Inequality Notation: Interval Notation:

26 26 INTRODUCTION TO FUNCTIONS Example 1 Domain and Range, Multiple Representation In the graph to the right, the line stretches to infinity in both directions. The domain is all real numbers. Inequality Notation: < x < Interval Notation: (, ) The range is all real numbers. Inequality Notation: < y < Interval Notation: (, ) We have the graph of a parabola opening up. The two ends stretch up and out to infinity. The domain is all real numbers. Inequality Notation: < x < Interval Notation: (, ) Notice that the graph goes no lower than 2 along the y-axis. The range is all values of y greater than or equal to 2. Inequality Notation: y 2 Interval Notation: [ 2, )

27 1.4 DOMAIN AND RANGE 27 Exercise 1 Domain and Range, Multiple Representations Find the domain and range of the functions below. Use proper notation for yor answers. a) Set of ordered pairs Domain: D(r)= {(7,8), (8,12), (11,21)} Range: b) Table of values n A(n) Domain: Range: c) Graph Domain of f (x) Inequality Notation: Interval Notation: Range of f (x) Inequality Notation: Interval Notation:

28 28 INTRODUCTION TO FUNCTIONS d) Graph Domain of g(x) Inequality Notation: Interval Notation: Range of g(x) Inequality Notation: Interval Notation: e) Graph Domain of h(x) Inequality notation: Interval Notation: Range of h(x) Inequality Notation: Interval Notation:

29 1.5 APPLICATIONS OF FUNCTIONS APPLICATIONS OF FUNCTIONS Criteria for a Good Graph: 1. Horizontal axis should be properly labeled with the name and units of the input quantity. 2. Vertical axis should be properly labeled with the name and units of the output quantity. 3. Use an appropriate scale. Start at or just below the lowest value. End at or just above the highest value. Scale the graph so the adjacent tick marks are equal distance apart. Axes must meet at (0, 0) called the origin. Use numbers that make sense for the given data set. Use // or a sawtooth mark between the origin and the first tick mark to show a skip in the range of values. 4. All points should be plotted correctly and the graph should be neat and uncluttered. Problem 1 : Class/Media - Understanding Applications of Functions Suppose that the cost to fill your 15-gallon gas tank is determined by the function C(g) = 3.29g where C is the output (cost in $) and g is the input (gallons of gas). a) What is the input quantity (including units) for this function? Name the smallest and largest possible input quantity (this is called the practical domain). b) What is the output quantity (including units) for this function? Name the smallest and largest possible output quantity (thisis called the practical range).

30 30 INTRODUCTION TO FUNCTIONS c) Identify the first and last ordered pairs that are on the graph (based on the information above). Include both ordered pairs and function notation. d) Draw a good graph of this function. Provide labels for your axes. The Practical Domain of a function is the set of all possible input values that are realistic for a given problem. The Practical Range of a function is the set of all possible output values that are realistic for a given problem.

31 1.5 APPLICATIONS OF FUNCTIONS 31 Example 1 Practical Domain and Range Let the function M(t) = 15t represent the distance you would travel bicycling t hours. Assume you can bike no more than 10 hours. Find the practical domain and practical range for this function. Solution. Begin by drawing an accurate graph of the situation. Try and determine the smallest and the largest input values then do the same thing for the output values. Practical Domain Practical Range In this situation, the input values you can use are related to biking and the input is time. You are told you can bike no more than 10 hours. You also cannot bike a negative number of hours but you can bike 0 hours. In this situation, the outputs represent distances traveled depending on how long you bike. Looking at the endpoints for practical domain, you can find the practical range as follows: M(0) M(t) M(10) Therefore, the practical domain is: 0 t 10 hours Thus, 0 M(t) 150 miles is the practical range.

32 32 INTRODUCTION TO FUNCTIONS Exercise 1 Applications of Functions A local towing company charges $3.25 per mile driven plus a base fee of $ They tow a maximum of 25 miles. a) Let C represent the total cost of any tow and x represent miles driven. Write a function that represents the total cost as a function of miles driven using correct function notation. b) Identify the practical domain of this function in the form: Minimum Miles Towed x Maximum Miles Towed Practical Domain: c) Identify the practical range of this function in the form: Minimum Cost C(x) Maximum Cost Practical Range: d) Use your function from part (a) to find C(15). Write your answer as an ordered pair and then explain its meaning in a complete sentence. e) Use your function from part (a) to determine the value of x when C(x) = 30. Write your answer as an ordered pair and then explain its meaning in a complete sentence.

33 1.5 APPLICATIONS OF FUNCTIONS 33 Exercise 2 Applications of Functions The value, V (in dollars) of a washer/dryer set decreases as a function of time, t (in years). The function V(t)= 100t models this situation. You own the washer/dryer set for 12 years. a) Identify the input quantity (including units) and the input variable. b) Identify the output quantity (including units) and the output variable. c) Fill in the table below. t V(t) d) Draw a GOOD graph of this function below. Provide labels for your axes. Plot and label the ordered pairs from (c).

34 34 INTRODUCTION TO FUNCTIONS e) How old is a washer/dryer set that is worth $400? Show your work. f) How much would the washer/dryer set be worth after two years? Show your work. g) What is the practical domain for V(t)? Inequality Notation: Interval Notation: h) What is the practical range for V(t)? Inequality Notation: Interval Notation:

35 PRACTICE PROBLEMS 35 Section 1.1: What is a Function? PRACTICE PROBLEMS 1. The table below gives the distance, D, in kilometers, of a GPS satellite from Earth t minutes after being launched. t = Time (in minutes) D = Distance (in km) , , , ,200 a) Identify the input quantity (include units). b) Identify the input variable. c) Identify the output quantity (include units). d) Identify the output variable. e) Write the data as a set of ordered pairs. f) In a complete sentence, interpret the meaning of the ordered pair (40, 9452). g) Is the distance of the satellite a function of time? Why or why not?

36 36 INTRODUCTION TO FUNCTIONS 2. The table below gives the number of gene copies, G, t minutes after observation. t = Time (in minutes) G = number of gene copies a) Identify the input quantity (include units). b) Identify the input variable. c) Identify the output quantity (include units). d) Identify the output variable. e) Write the data as a set of ordered pairs. f) In a complete sentence, interpret the meaning of the ordered pair (6, 208). g) Is the number of gene copies a function of time? Why or why not

37 PRACTICE PROBLEMS The table below gives the number of homework problems, H, that Tara has completed t minutes after she began her homework. t = Time (in minutes) H = Homework Problems Completed a) Identify the input quantity (include units). b) Identify the input variable. c) Identify the output quantity (include units). d) Identify the output variable. e) Write the data as a set of ordered pairs. f) In a complete sentence, interpret the meaning of the ordered pair (40, 15). g) Is the number of homework problems completed a function of time? Why or why not?

38 38 INTRODUCTION TO FUNCTIONS 4. The table below gives the number of hotdogs, H, eaten by a competitive hotdog eater, t minutes after the start of the competition. t = Time H = number of (in minutes) Hotdogs Eaten a) Identify the input quantity (include units). b) Identify the input variable. c) Identify the output quantity (include units). d) Identify the output variable. e) Write the data as a set of ordered pairs. f) In a complete sentence, interpret the meaning of the ordered pair (7, 50). g) Is the number of hotdogs eaten a function of time? Why or why not?

39 PRACTICE PROBLEMS 39 Section 1.2: Multiple Representations of Function 5. Determine whether the following sets of ordered pairs represent a functional relationship or not. Justify your answer. a) R= {(2,4), (3,8), ( 2,6)} b) T = {(3, 2), (4, 1), (5,8), (3, 2)} c) L = {(3, 5), (1, 2), (2, 2), (3,5)} d) A= {( 5, 5), (6, 5), (7, 5)} e) F = {(2, 3), (2,3), (4,8)}

40 40 INTRODUCTION TO FUNCTIONS 6. Determine whether the following tables of values represent a functional relationship or not. Justify your answer. a) b) c) x f (x) x g(x) x h(x) d) e) r h(r) s m(s) f) g) t p(t) s r(s)

41 PRACTICE PROBLEMS Determine if the following graphs represent a functional relationship. Justify your answer. a) b) c) d) e) f)

42 42 INTRODUCTION TO FUNCTIONS 8. Determine whether the following scenarios represent a function or not. Explain your choice for each. Remember when the word function is used here, it is in a purely mathematical sense, not in an everyday sense. a) Is a person's height a function of their age? b) Is a person's age a function of their date of birth? c) Is the growth of a tree a function of the monthly rainfall? d) John says that the time he will spend on vacation will be determined by the number of overtime hours he works on his job. Is it ture that his vacation time is a function of his overtime hours? e) Sara says that the size of the tomatoes she grows will be determined by the weather. Is it true that the size of her tomato crop is a function of the weather?

43 PRACTICE PROBLEMS Determine if the functions represented by the graphs below are increasing, decreasing or constant. a) b) c) d) e) f)

44 44 INTRODUCTION TO FUNCTIONS 10. Determine whether the following functions represented by the tables below are increasing, decreasing or constant. a) b) c) x f (x) t s(t) x g(x) d) x h(x) e) t p(t) f) s r(s)

45 PRACTICE PROBLEMS 45 Section 1.3: Function Evaluation 11. Given the function f (x)= x + 6, evaluate each of the following: a) f (2) b) f ( 1) c) f (0) d) f (2n) e) f ( 1 2 a ) f) f (x 3) 12. Given the function s(t)= 14 2t, evaluate each of the following: a) s( 3) b) s(4) c) s(0) d) s(3a) e) s ( 3 2 c ) f) s(t + 4)

46 46 INTRODUCTION TO FUNCTIONS 13. Given the function h(c)= 2c 2 3c+ 4, evaluate each of the following: a) h( 2) b) h(3) c) h(0) d) h( c) e) h(c 1) f) h(3x +2) 14. Given the function g(x)= x 2 + 3x, evaluate each of the following: a) g( 3) b) g( x) c) g(1 x)

47 PRACTICE PROBLEMS Given f (x) = 3x 6, determine each of the following. Also determine if you are given an input or an output value and whether you are finding an input or an output value. Then write the result as an ordered pair. a) Find f (2) b) Find x if f (x)= 3 Given input or output? Given input or output? Finding input or output? Finding input or output? Ordered Pair: Ordered Pair: c) Find f ( 4) d) Find x if f (x)= 12 Given input or output? Given input or output? Finding input or output? Finding input or output? Ordered Pair: Ordered Pair:

48 48 INTRODUCTION TO FUNCTIONS 16. Given g(x) = 3 2 x 1, determine each of the following. Also determine if you are given an 2 input or an output value and whether you are finding an input or an output value. Then write the result as an ordered pair. Keep answers as fraction, when appropriate. a) Find g ( 1 4) b) Find x if g(x)= 3 Given input or output? Given input or output? Finding input or output? Finding input or output? Ordered Pair: Ordered Pair: c) Find g( 8) d) Find x if g(x)= 7 2 Given input or output? Given input or output? Finding input or output? Finding input or output? Ordered Pair: Ordered Pair: 17. Use the table below to find the function values. a) k(7) t k(t) b) k( 3) c) k( 8)

49 PRACTICE PROBLEMS Use the table below to determine each of the following. Also determine if you are given an input or an output value and whether you are finding an input or an output value. Then write the result as an ordered pair. x f (x) a) Find x if f (x)= 2 b) Find f ( 4) Given input or output? Given input or output? Finding input or output? Finding input or output? Ordered Pair: Ordered Pair: c) Find x if f (x)= 6 d) Find f ( 2) Given input or output? Given input or output? Finding input or output? Finding input or output? Ordered Pair: Ordered Pair:

50 50 INTRODUCTION TO FUNCTIONS 19. Use the graph of f (x) below to determine each of the following. Also determine if you are given an input or an output value and whether you are finding an input or an output value. Then write the result as an ordered pair. a) Find all x-values if f (x) = 3 b) Find f ( 2) Given input or output? Given input or output? Finding input or output? Finding input or output? Ordered Pair: Ordered Pair: c) Find all x-values if f (x) = 2 d) Find f (3) Given input or output? Given input or output? Finding input or output? Finding input or output? Ordered Pair: Ordered Pair:

51 PRACTICE PROBLEMS Use the graph of g(x) below to determine each of the following. Also determine if you are given an input or an output value and whether you are finding an input or an output value. Then write the result as an ordered pair. a) Find all x-values where g(x)= 6 b) Find g(1) Given input or output? Given input or output? Finding input or output? Finding input or output? Ordered Pair: Ordered Pair: c) Find all x-values where g(x)= 0 d) Find g(0) Given input or output? Given input or output? Finding input or output? Finding input or output? Ordered Pair: Ordered Pair:

52 52 INTRODUCTION TO FUNCTIONS 21. Consider the function h(x) = 2x 3. a) Complete the table below b) Sketch the graph of h(x) =2x 3 x h(x) 22. Consider the function f (x) = 3x + 4. a) Complete the table below b) Sketch the graph of h(x) = 3x +4 x f (x)

53 PRACTICE PROBLEMS 53 Section 1.4: Domain and Range 23. For each set of oredered pairs, determine the domain and range. a) g= {(3, 2), (5, 1), (7,8), (9, 2), (11,4), (13, 2)} Domain: Range: b) f = {( 2, 5), ( 1, 5), (0, 5), (1, 5)} Domain: Range: c) h= {( 3,2), (1, 5), (0, 3), (4, 2)} Domain: Range: 24. For each table of values, determine the domain and range of the function. a) b) x f (x) x g(x) Domain: Range: Domain: Range: c) t H(t) Domain: Range:

54 54 INTRODUCTION TO FUNCTIONS 25. For each graph, determine the domain and range of the function. a) b) Domain: Inequality Notation: Interval Notation: Range: Inequality Notation: Interval Notation: Domain: Inequality Notation: Interval Notation: Range: Inequality Notation: Interval Notation: c) d) Domain: Inequality Notation: Interval Notation: Range: Inequality Notation: Interval Notation: Domain: Inequality Notation: Interval Notation: Range: Inequality Notation: Interval Notation:

55 PRACTICE PROBLEMS 55 Section 1.5: Applications of Functions 26. A local window washing company charges $0.50 per window plus a base fee of $20.00 per appointment. They can wash a minimum of 5 windows and a maximum of 200 windows per appointment. a) Let C represent the total cost of an appointment and w represent the number of windows washed. Using correct and formal function notation, write the function that represents total cost as a function of windows washed. b) Identify the practical domain of this function by filling in the blanks below. Minimum Windows Washed w Maximum Windows Washed Practical Domain: c) Identify the practical range of this function by filling in the blanks below. Minimum Cost C(w) Maximum Cost Practical Range: d) Complete the table below: w C(w) e) Find the value of C(50). Circle the appropriate column in the table. Write a sentence explaining the meaning of your answer. f) Use your function from part (a) to determine the value of w when C(w)=60. Set up the equation, C(w) = 60 then solve for the value of w. Write a sentence explaining the meaning of your answer.

56 56 INTRODUCTION TO FUNCTIONS 27. Suppose the number of pizzas you can make in an 8-hour day is determined by the function P(t)= 12t where P is the output (pizzas made) and t is the input (time in hours). a) What is the input quantity (including units) for this function? Name the smallest and largest possible input quantity then use this information to identify the practical domain. Input quantity (including units): Practical Domain: Inequality Notation: Interval Notation: b) What is the output quantity (including units) for this function? Name the smallest and largest possible output quantity then use this information to identify the practical range. Output quantity (including units): Practical Range: Inequality Notation: Interval Notation: c) Identify the first and last ordered pairs of the graph (based on the information above). Include both ordered pairs and function notation. d) Using the above information, show a good graph in the space below.

57 PRACTICE PROBLEMS 57 e) Find P(3) and interpret its meaning in the context of the problem. f) Find t so that P(t)= 70 and interpret its meaning in the context of the problem. 28. The life expectancy for males in the United States from the year 1900 until 2020 can be modeled by the function L(x) = 0.27x , where L is the life expectancy and x is the number of years since a) Which letter, L or x, is used for input? b) What does the input represent? (include units) c) Which letter, L or x, is used for output? d) What does the output represent? (include units)

58 58 INTRODUCTION TO FUNCTIONS e) Fill in the table and then draw a neat, labeled and accurate sketch of this graph in the space below. x 0 L(x) f) What is the practical domain of L(x)? Use proper inequality notation. g) What is the practical range of L(x)? Use proper inequality notation. h) What is the life expectancy of a man born in Iowa in 1950? (Round to the nearest whole number) i) If a man is expected to live to the age of 60, approximate the year he was born. (Round to the nearest whole year)

59 CHAPTER 1 ASSESSMENT 59 CHAPTER 1 ASSESSMENT 1. Let r(a)= 4 5a. Write each answer using function notation and as an ordered pair. a) Determine r( 2). b) For what value of a is r(a)= 1? 2. The graph of f (x) is shown below. Use inequality notation to answer the questions. a) Give the domain of f (x). b) Give the range of f (x). c) Find f (0). d) f (x) =0whenx =? 3. Consider the following table of values. Fill in the blanks below and identify the corresponding ordered pairs. a) Find g(1) x g(x) b) g(x)= 1 when x =? c) g(x) = 2 when x =?

60 60 INTRODUCTION TO FUNCTIONS 4. Given the function m(t)= 5 3t, evaluate each of the following: a) m( 2) b) m ( 2 3 t ) c) m(x 4) 5. Given the function h(c)= 4+ c c 2, evaluate each of the following: a) h( 2) b) h(c 1) c) h(x + 2)

61 CHAPTER 2 FUNCTIONS AND FUNCTION OPERATIONS As we continue to work with more complex functions, it is important that we are comfortale with Function Notation, operations on functions and operations involving more than one function. In this chapter, we study proper usage of the function notation and then spend time learning how to add, subtact, multiply and divide functions, both algebraically and when the functions are represented with a table or graph. Finally, we take a look at a couple of real world examples that involve operations on functions. Chapter Topics: Section Combining Functions Basic Operations: Addition, Subtraction, Multiplication and Division Multiplication Property of Exponents Division Property of Exponents Negative Exponents Operations on Functions in Table Form Operations on Functions in Graph Form Section Applicaitons of Function Operations Cost, Revenue and Profit 61

62 62 FUNCTIONS AND FUNCTION OPERATIONS Component Required? Yes or No Comments Due Score Mini-Lesson Homework Quiz Test Practice Problems Chapter Assessment

63 2.1 COMBINING FUNCTIONS COMBINING FUNCTIONS Function notation can be expanded to include notation for the different ways we combine functions as described below. Basic Mathematical Operations on Functions The basic mathematical operations are: addition, subtraction, multiplication and division. When working with function notation, these operations will look like this: Addition Subtraction Multiplication Division f (x) +g(x) f (x) g(x) f (x) g(x) f (x), g(x) 0 g(x) Many of the problems in this section are problems you may already know how to do. You will just need to get used to some new notation. Example 1 Adding and Subtracting Functions Given f (x)= 2x 2 + 3x 5 and g(x)= x 2 + 5x + 1, determine each of the following. a) f (x)+ g(x) b) f (x) g(x) c) f (1) g( 1) Solution. a) f (x)+ g(x) f (x) +g(x) = (2x 2 +3x 5) +( x 2 +5x +1) = 2x 2 +3x 5 x 2 +5x +1 = x 2 +8x 4 b) f (x) g(x) f (x) g(x) = (2x 2 +3x 5) ( x 2 +5x +1) = 2x 2 +3x 5+x 2 5x 1 = 3x 2 2x 6 c) f (1) g( 1) f (1) g(1) = [2(1) 2 +3(1) 5] [ ( 1) 2 +5( 1) +1] = [2(1) +3(1) 5] [ (1) +5( 1) +1] = [2 +3 5] [ 1 5+1] = [0] [ 5] = 5

64 64 FUNCTIONS AND FUNCTION OPERATIONS Problem 1 : Class/ Media - Adding and Subtracting Functions Given f (x)= 3x 2 + 2x 1 and g(x)= x 2 + 2x + 5, determine each of the following. a) f (x)+ g(x) b) f (x) 2g(x) Exercise 1 Adding and Subtracting Functions Given f (x)= x and g(x)= x 2 + 1, determine each of the following. Show complete work. a) f ( 2)+ g(2) b) 5f (x) 2g(x) c) 3f (2) g( 2) MULTIPLICATION PROPERTY OF EXPONENTS Let m and n be rational numbers and let a be any real number, then a m.a n =a m+n

65 2.1 COMBINING FUNCTIONS 65 Example 2 Function Multiplication For each of the following, determine f (x) g(x) a) f (x)= 8x 4 and g(x)= 5x 3 b) f (x)= 3x and g(x)= 4x 2 x + 8 c) f (x)= 3x 2 and g(x)= 2x 5 Solution. a) f (x)= 8x 4 and g(x)= 5x 3 f (x) g(x) = ( 8x 4 ) (5x 3 ) = ( 8) (5)(x 4 ) (x 3 ) = 40x 7 b) f (x)= 3x and g(x)= 4x 2 x + 8 f (x) g(x) = ( 3x) (4x 2 x + 8) = ( 3x)(4x 2 ) +( 3x)( x) +( 3x)(8) = 12x 3 +3x 2 24x c) f (x)= 3x 2 and g(x)= 2x 5 f (x) g(x) = (3x + 2)(2x 5) = (3x)(2x 5)+ (2)(2x 5) = (3x)(2x)+ (3x)( 5)+ (2)(2x)+ (2)( 5) = 6x 2 15x + 4x 10 = 6x 2 11x 10 Problem 2 : Class/Media - Function Multiplication Given f (x)= 3x + 2 and g(x)= 2x 2 + 3x + 1, determine f (x) g(x)

66 66 FUNCTIONS AND FUNCTION OPERATIONS Exercise 2 Function Multiplication For each of the following, determine f (x) g(x) a) f (x)= 3x 2 and g(x)= 3x + 2 b) f (x)= 2x 2 and g(x)= x 3 4x + 5 c) f (x)= 4x 3 and g(x)= 6x DIVISION PROPERTY OF EXPONENTS Let m and n be rational numbers and let a be any real number, then am a n = am n, where a 0. NEGATIVE EXPONENTS If a 0 and n is a rational number, then a n = 1 an, where a 0.

67 2.1 COMBINING FUNCTIONS 67 Example 3 Function Division For each of the following, find f (x). Use only positive exponents in your final answer. g(x) a) f (x)= 15x 7 and g(x)= 3x 2 b) f (x)= 4x 5 and g(x)= 2x 8 Solution. a) Using Division Property of Exponents: f (x) g(x) = 15x7 3x 2 = 15 3 x7 2 = 5x 5 OR By Expanding Factors: f (x) g(x) = 15x 7 3x 2 1 = 5 3 x 1 x 1 x x x x x 3 x x = 5x 5 b) Using Division Property of Exponents: or By Expanding Factors: f (x) g(x) = 4x5 2x 8 = 4 2 x5 8 f (x) g(x) = 4x5 2x 8 = 2x 3 = 2 x 3 1 = 2 2 x 1 x 1 x 1 x 1 x 1 2 x x x x x x x x = 2 x 3

68 68 FUNCTIONS AND FUNCTION OPERATIONS DIVISION OF FUNCTIONS: When dividing two functions p(x) where p(x) and q(x) are polynomial function such that q(x) 0: q(x) If q(x) is a monomial, we perform the division of each term of p(x) by q(x). If q(x) is not a monomial, we perform the long division algorithm. Example 4 Dividing Monomials Determine f (x) g(x) given f (x)=4x2 +5x 3 and g(x)=2x 2. Use only positive exponents in your final answer. Solution. Since the denominator is a monomial, divide each term in the numerator by the denominator. 4x 2 + 5x 3 2x 2 = 4x2 2x 2 + 5x 2x 2 3 2x 2 = x 3 x 2 Problem 3 : Class/Media - Dividing Monomials For each of the following, determine f (x). Use only positive exponents in your final answer. g(x) a) f (x)= 12x 4 8x 3 2x 2 + 4x and g(x)= 4x b) f (x)= 5x x 2 25x and g(x)= 5x 3

69 2.1 COMBINING FUNCTIONS 69 Exercise 3 Dividing Monomials For each of the following, determine f (x). Use only positive exponents in your final answer. g(x) a) f (x)= 8x 3 6x 2 + 4x 12 b) f (x) = 9x 3 +6x 2 3x +12 and g(x)= 4x and g(x) = 3x 2 Example 5 Long Division Given f (x)= x 3 6x 2 5 and g(x)= x 2. Determine f (x) g(x). Solution. Since the denominator is a binomial and not a monomial, we will perform long division. STEP 1: STEP 2: STEP 3: STEP 4: Rewrite the terms in the numerator and denominator in descending order, putting a coefficient of 0 for the missing term: x3 6x 2 +0x 5 x 2 Divide the first term of the dividend by the first term of the divisor. x 2 x 2 )x 3 6x 2 + 0x + 5 Multiply the divisor by the result just obtained. x 2 x 2 ) x 3 6x 2 + 0x + 5 x 3 2x 2 Subtract like terms and bring down the next term. x 2 x 2 ) x 3 6x 2 + 0x + 5 (x 3 2x 2 ) 4x 2 + 0x

70 70 FUNCTIONS AND FUNCTION OPERATIONS STEP 5: Repeat steps 2-3. x 2 4x x 2 ) x 3 6x 2 + 0x + 5 (x 3 2x 2 ) 4x 2 + 0x 4x 2 + 8x STEP 6: Subtract like terms and bring down the next term. x 2 4x x 2 ) x 3 6x 2 + 0x + 5 (x 3 2x 2 ) 4x 2 + 0x ( 4x 2 + 8x) 8x + 5 STEP 7: Repeat steps 2-3. x 2 4x 8 x 2 ) x 3 6x 2 + 0x + 5 (x 3 2x 2 ) 4x 2 + 0x ( 4x 2 + 8x) 8x + 5 8x + 16 STEP 8: Subtract like terms and bring down the next term x 2 4x 8 x 2 ) x 3 6x 2 + 0x + 5 (x 3 2x 2 ) 4x 2 + 0x ( 4x 2 + 8x) 8x + 5 ( 8x + 16) 11 (remainder) This is the long division algorithm for polynomials. Therefore, x3 6x 2 +5 x 2 =x 2 4x 8 11 x 2.

71 2.1 COMBINING FUNCTIONS 71 Problem 4 : Class/Media - Long Division Divide the following using long division. 4x a) 3 +2x 2 2x 3 2x +1 b) (2x 3 +3x 2 +27) (x +3) Exercise 4 Long Division Divide the following using long division. x a) 3 x 2 2x +6 x 2 b) (9x 3 7x + 2) (3x + 2)

72 72 FUNCTIONS AND FUNCTION OPERATIONS Functions can be presented in multiple ways including: equations, data sets, graphs and applications. If you understand function notation, then the process for working with functions is the same no matter how the information is presented. Problem 5 : Class/Media - Working with Functions in Table Form x f (x) x g(x) Functions f (x) and g(x) are defined in the tables above. Find the following. a) f (1) b) g(11) c) f (0)+ g(0) d) g(5) f (8) e) f (0) g(3) f) g(9) f (5) Exercise 5 Working with Functions in Table Form x f (x) x g(x) Given the above two tables. Complete the tablebelow. Show your workin the table cell for each column. The first one is done for you. x f (x)+ g(x) f (0) +g(0) = = 10

73 2.1 COMBINING FUNCTIONS 73 Remember that graphs are just infinite sets of ordered pairs and if you do a little work ahead of time (as in the example below), then the graphing problems are a lot easier to work with. Exercise 6 Working with Functions in Graph Form Complete the tables of ordered pairs using the graphs on the left. Assume integer answers. Then use the information to help answer the problems below. One ordered pair for each function has been completed. f ( 6, 2 ) ( 5, ) ( 4, ) ( 3, ) ( 2, ) ( 1, ) ( 0, ) (1, ) ( 2, ) ( 3, ) ( 4, ) ( 5, ) ( 6, ) g ( 6, ) ( 5, 1 ) ( 4, ) ( 3, ) ( 2, ) ( 1, ) ( 0, ) ( 1, ) ( 2, ) ( 3, ) ( 4, ) ( 5, ) ( 6, ) a) g(4)= b) f (2) = c) g(0)= d) f ( 6) = e) If f (x)= 0, x = f) If g(x) =0, x = g) f ( 1)+ g( 1)= h) If g(x) = 4, x = i) f ( 1)+ g( 1)= j) g( 6) f ( 6) = k) f (1)+ g( 2)= l) g(6) f ( 1) =

74 74 FUNCTIONS AND FUNCTION OPERATIONS 2.2. APPLICATIONS OF FUNCTION OPERATIONS One of the classic applications of function operations comes from the business. The Profit Function, P(x) is obtained by subtracting the Cost Function, C(x), from the Revenue Function, R(x) as shown below. Profit = Revenue Cost P(x)=R(x) C(x) where P(x) is the Profit Function, R(x) is the Revenue Function, and C(x) is the Cost Function. Problem 1 : Class/Media - Cost, Revenue, Profit A local courier service estimates its monthly operating costs to be $1500 plus $0.85 per delivery. The service generates revenue of $6 for each delivery. Let x = the number of deliveries in a given month. a) Write a function, C(x), to represent the monthly costs for making x deliveries per month. b) Write a function, R(x), to represent the revenue for making x deliveries per month. c) Write a function, P(x), to represent the monthly profit for making x deliveries per month. d) Determine algebraically the break-even point for the function P(x) by solving the equation P(x)= 0 for x. How many deliveries must you make each month to begin making money? Round to the nearest whole number. Show complete work. Write your final answer in a complete sentence.

75 2.2 APPLICATIONS OF FUNCTION OPERATIONS 75 Exercise 1 Cost, Revenue, Profit February is a busy time at Charlie's Chocolate Shoppe! During the week before Valentine's Day, Charlie advertises that his chocolates will be selling for $1.80 a piece (instead of the usual $2.00 each). The fixed weekly cost to run the Chocolate Shoppe totals $450 and each chocolate costs about $0.60 to produce. Charlie estimates he can produce up to 3,000 chocolates in one week. a) Write a function, C(n), to represent Charlie's total cost if he makes n chocolates during the week before Valentine's Day. b) Write a function, R(n), to represent the revenue from the sale of n chocolates during the week before Valentine's Day. c) Write a function, P(n), to represent Charlie's profit from selling n chocolates during the week before Valentine's Day. Show complete work. d) Interpret the meaning of the statement P(300)= 90.

76 76 FUNCTIONS AND FUNCTION OPERATIONS e) Determine the practical domain and practical range for P(n). Then use the information to graph P(n). Practical Domain: Practical Range f) How many chocolates must Charlie sell in order to break even? Show complete work. Write your final answer in a complete sentence. Mark the break even point on the graph above.

77 PRACTICE PROBLEMS 77 Section 2.1: Combining Functions PRACTICE PROBLEMS 1. Let f (x)=2x 3 and g(x)=4x 2 +6x 7. Determine the following and simplify your result. a) f (4)+ g(2) b) g(x) 2f (x) c) f (2) g(1) d) g(x) f (x) 2. Let f (x)=15x 2 3x +4 and g(x)= 3x 2. Determine the following and simplify your result. a) 2f (x)+ g(x) b) 4f (x) 5g(x) c) f (x) g(x) d) f (x) g(x)

78 78 FUNCTIONS AND FUNCTION OPERATIONS 3. Given f (x)= 6x 2 4x + 2 and g(x)= 3x + 1. Find the following and simplify your result. a) f (x) g(x) b) f (4) +g( 1) c) f (x) g(x) d) f (x) g(x) 4. Determine f (x) and simplify your answers. Use only positive exponents in your answer. g(x) a) f (x)= 30x 4 5x 3 20x 15 and g(x)= 5x 2 b) f (x)= x 4 3x 2 x 2 and g(x)= x 2 c) f (x)= 8x 4 + 4x 2 3x + 1 and g(x)= 2x 2 x + 2

79 PRACTICE PROBLEMS Use the tables of the functions below to determine the following. x m(x) x n(x) a) m(2) + n(2) b) n( 2) m( 2) c) m( 1) n(0) d) n(1) m(1) 6. Functions h(x) and k(x) are defined in the tables below. Use those tables to evaluate the problems below. x h(x) x k(x) a) h(5) b) k(5) c) h(0) + k(15) d) k(11) k(8) e) h(0) k(4) f) k(9) h(1)

80 80 FUNCTIONS AND FUNCTION OPERATIONS 7. Use the graph to determine each of the following. Assume integer answers. a) f (0)+ g(0) b) f (1) g(3) c) f ( 2) g(5) d) g(2) f ( 1) 8. Functions f and g are defined below. Use those functions to evaluate the problems below. f = {( 3,4), ( 2,6), ( 1,8), (0,6), (1, 2)} g ={( 1, 8),(0, 2),(4, 3),(8, 4)} a) f ( 2)+ g(0) b) f (1) g(4) c) f (0) g(0) d) f ( 1) g(8)

81 PRACTICE PROBLEMS 81 Section 2.2: Applications of Function Operations 9. The function E(n) represents Ellen's budgeted monthly expenses for the first half of the year In the table, n=1 represents January 2016, n=2 represents February 2016, and so on. n E(n) $2263 $2480 $2890 $2263 $2352 $2550 The function L(n) shown in the table below represents Ellen's monthly income for the first half of the year In the table, n=1 represents January 2016, n=2 represents February 2016, and so on. n L(n) $2850 $2850 $2850 $2850 $2850 $2850 a) At the end of each month, Ellen puts any extra money she has into a saving account. The function S(n) represents the amount of money she puts into her savings account each month. Using the information above, complete the following table for the function S(n). n S(n) b) Her goal is to save enough money to take a trip to Hawaii in July, She estimates that the trip will cost $2000. Will she be able to save up enough money to go to Hawaii in July? If so, how much extra spending money will she have while she is there? If not, how much more does she need to earn?

82 82 FUNCTIONS AND FUNCTION OPERATIONS 10. Maria and Todd are organizing the 20th year reunion for their high school. The high school alumni asosciation has given them $1000 for the event. They talk to the local caterer and find out the following: It will cost $15 per person plus a $50 setup fee to provide food for the event. It will cost $3 per person plus an $80 setup fee to rent the Meeting Hall at the local Holiday Motel. To help determine the costs, they come up with the following functions where x represents the number of people: The cost for food is $50 + $15 per person or F(x)= 15x + 50 The cost for the Hall is $80 + $3 per person or H(x)= 3x + 80 In addition, they decide to charge each person $5 to get in the door. This can be modeled by the following function: Income for the event is $1000 from the alumni + $5 per person or I(x)= 5x Given this information, answer the following questions. Show how you use the functions to calculate the answers and give your final answers in complete sentences. If 400 people attend the event, a) How much will food cost? b) How much will the Meeting Hall rental cost? c) How much will food AND the Meeting Hall rental cost? Show how you use the functions to calculate this. (Hint: F(400) +H(400)) d) The final bill for the event is found by subtracting the costs from the income. What would the final bill for the event be? e) Challenge: How many people can attend if the costs have to equal the income?

83 PRACTICE PROBLEMS Leonard has started a new business making cartoon bedspreads. His monthly expenses are $1322. Each bedspread costs $8.50 to produce. a) Complete the table below showing Leonard's business costs, C(n) as a function of the number of bedspreads, n, he makes. n C(n) b) Leonard is selling each bedspread for $ Complete the table below showing Leonard's revenue, R(n), as a function of the number of bedspreads, n, he sells. n R(n) c) The profit from Leonard's business can be found by subtracting the cost function from the revenue function. Complete the table below showing Leonard's profit, P(n), as a function of the number of bedspreads, n, he sells. n P(n) d) Using the table from part (c), make a rough estimate for the number of bedspreads Leonard needs to sell for his business to break even. (Note: Breaking even means profit = 0) e) Using the information from parts (a) through (c), create the algebraic functions for C(n), R(n), and P(n). C(n)= R(n)= P(n)= f) Using your formula for profit, P, determine the exact number of bedspreads Leonard needs to sell for his business to break even. (Note: Breaking even means profit = 0)

84 84 FUNCTIONS AND FUNCTION OPERATIONS CHAPTER 2 ASSESSMENT 1. Simplify each of the following, if possible. a) 2n 5 + 3n 5 b) 12n 5 3n 5 c) 2n 3 3n 5 d) 12n 5 3n 3 2. The functions A(x) and B(x) are defined by the following tables. x A(x) x B(x) Determine the values for each of the following. a) B(3) b) A(8) c) A( 3) + B(11) d) A(10) B(15) e) A(4) B(4) f) A(0) B(9)

85 CHAPTER 2 ASSESSMENT Let p(x)= x 2 + 2x 3 and r(x)= x 5. Deteremine each of the following. Show all work. a) p(x) 2r(x) b) p(0) r(0) 4. Let f (x)= x 2 +3x 4 and g(x)=2x +5. Determine each of the following. Show all work. a) f (3)+ g ( 1 2) b) g(2) f ( 1) 5. Let h(x)=6x 2 x 15 and k(x)=2x 3. Determine each of the following. Show all work. a) h(x) k(x) b) h(x) k(x)

86

87 CHAPTER 3 LINEAR EQUATIONS AND FUNCTIONS The first function that we are going to investigate is Linear Function. This is a good place to start because with linear functions, the average rate of change is constant and no exponents higher than one are involved. Before we begin working with linear functions, we will review the characteristics of linear equations and operations on linear equations. Chapter Topics: Section Linear Equations and Functions Slope Slope-Intercept Form, y = mx+b y-intercepts x-intercepts Section Graphs of Linear Functions Graph by Plotting Points Graph Using Slope Graph Using Intercepts Section Horizontal and Vertical Lines Equations of Horizontal and Vertical Lines Graphs of Horizontal and Vertical Lines Section Writing the Equation of a Line Linear Equations from Graphs Applications of Linear Equations Section Systems of Linear Equations Recognizing Systems from Graphs Solving Systems of Linear Equations Applications of Systems of Linear Equations 87

88 88 LINEAR EQUATIONS AND FUNCTIONS Component Required? Yes or No Comments Due Score Mini-Lesson Homework Quiz Test Practice Problems Chapter Assessment

89 3.1 LINEAR EQUATIONS AND FUNCTIONS LINEAR EQUATIONS AND FUNCTIONS Linear equations should be familiar to students strarting Intermediate Algebra. The basics are covered here as a review of the important ideas and concepts that will be heavily utilized in the next sections. Slope Slope is a measure of steepness and direction of a given line. It is denoted by the letter m. Given any two points, (x 1, y 1 ) and (x 2, y 2 ), the slope of the line through the two points is determined by computing the following ratio: m = Change in Output Change in Input = y2 y1 Change in y = x 2 x 1 Change in x = y x Note. If the slope is positive, the line increases from left to right. If the slope is negative, the line decreases from left to right. If the slope is zero, then the line is horizontal (constant). If the slope is undefined, then the line is vertical. Example 1 Determine Slope of a Line Through Two Given Points Find the slope of the line through the given points. Then determine if the line is increasing, decreasing, horizontal or vertical. a) (2, 5) and ( 3, 4) b) ( 2, 4) and (4,8) c) (2,5) and (8, 5) Solution. a) (2, 5) and ( 3, 4) 4 ( 5) m = 3 (2) = = 9 5 = 9 5 Decreasing

90 90 LINEAR EQUATIONS AND FUNCTIONS b) ( 2, 4) and (4, 8) m = 8 ( 4) 4 ( 2) = = 12 6 = 2 Increasing c) (2,5) and (8, 5) m = = 0 6 = 0 Horizontal (Constant) Exercise 1 Determine Slope of a Line Through Two Given Points Find the slope of the line through the given points. Then determine if the line is increasing, decreasing, horizontal or vertical. a) (5, 2) and ( 3, 4) b) (6, 2) and (4, 6) Slope-Intercept Form for the Equation of a Line A LINEAR EQUATION is an equation that can bewritten in the form: y =mx +b, with slope, m and y-intercept (0, b) Using function notation, this equation of a line can be written as f (x) =mx +b

91 3.1 LINEAR EQUATIONS AND FUNCTIONS 91 y-intercept (or Vertical Intercept): (0, b) The y-intercept is the special ordered pair with coordinates (0, b). The input value is 0 and the resulting output is b. The y-intercept is often used to help when graphing a linear equation. It is the point of intersection between the line and the y-axis (or vertical axis). In practical applications, the y-intercept (or vertical intercept) helps to determine the initial output value. There are 3 main methods for finding the y-intercept of a linear equation or function. Method 1: Read the value of b from y =mx +b or f (x) =mx + b form. Method 2: Solve for y when x =0. Method 3: Evaluate f (0). Example 2 Determine y-intercept of a Linear Equation a) Find the y-intercept of the linear equation y = 2x 5. Solution. Using Method 1: The equation is written in the form y = mx+ b. Here, b= 5. Therefore, the y-intercept is (0, 5). Using Method 2: set x to 0 and solve for y. Therefore, the y-intercept is (0, 5). y = 2(0) 5 = 0 5 = 5 b) Find the y-intercept of the linear function f (x)= 2x 5. Solution. Using Method 3: Evaluate f (0). f (0) = 2(0) 5 = 0 5 = 5 f (0)= 5, therefore, the y-intercept is (0, 5).

92 92 LINEAR EQUATIONS AND FUNCTIONS Problem 1 : Class/Media - Determine Slope and y-intercept Complete the following table by rewriting the equation in function notation, determining the slope and behavior (increasing, decreasing or constant) and then writing the y-intercept as an ordered pair. Equation f (x) = mx + b Form Slope and Behavior y-intercept a) y = 2x +5 b) y = 2 x c) y = 3 4 x +2 d) y = 4x e) y = 6 f) y = x

93 3.1 LINEAR EQUATIONS AND FUNCTIONS 93 x-intercept (or Horizontal Intercept) : (a, 0) The x-intercept is the special ordered pair with coordinates (a, 0). The value a is the input value that results in an output of 0. The x-intercept is often used to help when graphing a linear equation. It is the point of intersection between the line and the x-axis (or horizontal axis). In practical applications, the x-intercept (or vertical intercept) helps to determine the final input value. There are 2 main methods for finding the x-intercept depending on what form the equation of the line is given. Method 1: From y =mx +b form, set y =0 and solve for x. Method 2: From f (x) =mx +b form, set f (x) =0 and solve for x. Example 3 Find the x-intercept of a Linear Equation a) Find the x-intercept of the linear equation y = 2x 5. Solution. Using Method 1: Replace the value of y with 0, then solve for the value of x. Therefore, the x-intercept is ( 5 2, 0 ). 0 = 2x 5 5 = 2x 5 2 = x b) Find the x-intercept of the linear function f (x)= 2x. Solution. Using Method 2: Set f (x)= 0 and solve for x. 0 = 2x 0 2 = x 0 = x Therefore, the x-intercept is (0, 0) which is also called the origin.

94 94 LINEAR EQUATIONS AND FUNCTIONS Problem 2 : Class/Media - Find the x-intercept For each of the following problems, determine the x-intercept, written as an ordered pair. a) y = 2x + 5 b) f (x)= 2 x c) g(x)= 3 x +2 d) y = 4x 4 e) h(x)= 6 f) y = x Exercise 2 Finding x- and y-intercepts of a Linear Equation/Function Complete the table below. Write intercepts as ordered pairs. Equation Slope and Behavior y-intercept x-intercept a) f (x) =6 4x b) y =3x c) y = 3 5 x 8

95 3.2 GRAPHS OF LINEAR EQUATIONS GRAPHS OF LINEAR EQUATIONS Problem 1 : Class/Media - Graphing a Linear Equation by Plotting Points Fill the table below and then graph f (x)= 2x + 6. x f(x) Ordered Pair Problem 2 : Class/Media - Using Slope and y-intercept to Graph a Linear Equation Given y = 2x + 6. Identify the slope and y-intercept. Use the information to draw an accurate graph. Then, identify at least two additional points on the line and label them on the graph.

96 96 LINEAR EQUATIONS AND FUNCTIONS Example 1 Using Intercepts to Graph a Linear Equation Graph the equation y = 2x + 6 by plotting the intercepts on the graph. Solution. y-intercept: This equation is written in the form y = mx+ b. Therefore, y-intercept is (0, 6). x-intercept: Set y to 0 and solve for x. Therefore, the x-intercept is (3,0). y = 2x = 2x = 2x 3 = 0 Plot and label the intercepts on the graph then draw your line through the intercepts.

97 3.2 GRAPHS OF LINEAR EQUATIONS 97 Exercise 1 Draw Graphs of Linear Equations Use the equation y = 3 x +6 for all parts of this problem. Label all plotted points. 2 a) Use the intercepts to graph of the line. Show your work. Plot and label the intercepts on the graph then draw your line through them. y-intercept: (, ) x-intercept: (, ) b) Use the y-intercept and slope to graph the line. Then identify at least two additional points on the line (not the intercepts) to confirm that the points are solutions to the equation. Label the two additional points on the graph. Slope = Two additional ordered pairs: (, ) (, ) NOTICE: Your graphs for parts (a) and (b) should look exactly the same.

98 98 LINEAR EQUATIONS AND FUNCTIONS 3.3. HORIZONTAL AND VERTICAL LINES Vertical Lines Equation: x =a x-intercept: (a, 0) y-intercept: none Slope: m is undefined Horizontal Lines Equation: y =b or f (x) =b x-intercept: none y-intercept: (0, b) Slope: m =0 Problem 1 : Class/Media - Graphing Horizontal and Vertical Lines a) Use the grid below to graph the equation y = 2. Identify the slope and intercepts. b) Use the grid below to graph the equation x = 5. Identify the slope and intercepts.

99 3.3 HORIZONTAL AND VERTICAL LINES 99

100 100 LINEAR EQUATIONS AND FUNCTIONS Exercise 1 Horizontal and Vertical Lines a) Given the ordered pair (2, 3) Sketch the graph of the vertical line through the given point. Write the equation of the veritcal line through the given point. Use function notation, if possible. Identify the slope of the line. What is the y-intercept? What is the x-intercept? b) Given the ordered pair (2, 3) Sketch the graph of the horizontal line through the given point. Write the equation of the veritcal line through the given point. Use function notation, if possible. Identify the slope of the line. What is the y-intercept? What is the x-intercept?

101 3.4 WRITING THE EQUATION OF A LINE WRITING THE EQUATION OF A LINE Writing Equations of Lines Critical to a thorough understanding of linear equations and functions is the ability to write the equation of a line given different pieces of information. The following process will work for almost every situation, unless you have a vertical line. Step 1: Determine the value of the slope, m. Step 2: Determine the coordinates of one ordered pair, (x 1, y 1 ). Step 3: Use the point-slope form, y y 1 =m(x x 1 ) to find the equation of the line. Step 4: Rewrite the resulting equation in slope-intercept form, y =mx +b. Step 5: When appropriate, rewrite the equation in function notation: f (x) =mx +b. Problem 1 : Class/Media - Writing Equations of Lines For each of the following, find the equation of the line that meets the following criteria. Write your final answer in function notation, if possible. a) Line has slope m= 4 and passes through the point (0,3) b) Line passes through the points (0, 2) and (1, 5)

102 102 LINEAR EQUATIONS AND FUNCTIONS c) Line passes through the points ( 2, 3) and (4, 9) d) Horizontal line passing through the point ( 3,5) e) Vertical line passing through ( 3,5) Example 1 Writing Equations of Lines Write the equation of the line that satisfies each set of conditions. a) Line contains the points ( 3,5) and (1, 0) Solution. Slope: Use the ordered pairs ( 3,5) and (1, 0) to compute the slope. m = ( 3) = 5 4 or 5 4 Ordered Pair: Choose one ordered pair: (1,0) Point-slope Form: Put the slope and ordered pair into the point-slope formula then solve for y. y y 1 = m(x x 1 ) y 0 = 5 (x 1) 4 y = 5 4 x Equation: Equation of line in slope-intercept form: y = 5 4 x Function Notation: f (x)= 5 4 x + 5 4

103 3.4 WRITING THE EQUATION OF A LINE 103 b) Line has slope m= 1 and goes through the point (4, 1) 2 Solution. Slope: Slope is give as m= 1 2 Ordered Pair: Ordered pair is (4, 1) Point-slope Form: Put the slope and ordered pair into the point-slope formula then solve for y. y y 1 = m(x x 1 ) y ( 1) = 1 (x 4) 2 y + 1 = 1 2 x 2 y = 1 2 x 3 Equation: Equation of line in slope-intercept form: y = 1 2 x 3 Function Notation: f (x)= 1 2 x 3 c) Vertical line going through the point (2, 6) Solution. Slope: Vertical line has undefined slope. Ordered Pair: Ordered pair is (2, 6) Point-slope Form: We cannot use point-slope form for vertical lines because the slope is undefined. Equation: Equation of the veritcal line is x = 2 Function Notation: We cannot write the equation of a vertical line in function notation. A vertical line is not a function and does not satisfy the condition of a function because one of its input has multiple outputs.

104 104 LINEAR EQUATIONS AND FUNCTIONS d) A line has the following graph: Solution. Slope: Identify two ordered pairs from the graph and use them to determine the slope. We will use (5,0) and (0, 3) to compute the slope. 3 (0) m = 0 (5) = 3 5 = 3 5 Ordered Pair: Choose one ordered pair that is on the line: (5,0) Point-slope Form: Put the slope and ordered pair into the point-slope formula then solve for y. y y 1 = m(x x 1 ) y 0 = 3 (x 5) 5 y = 3 5 x 3 Equation: Equation of line in slope-intercept form: y = 3 5 x 3 Function Notation: f (x)= 3 5 x 3

105 3.4 WRITING THE EQUATION OF A LINE 105 Exercise 1 Writing Equations of Lines A line goes through the points (1,4) and (3, 2). Answer the following questions. a) Find the equation of the line passing through the given points. Write your answer in function notation. Show complete work. b) What is the y-intercept of this line? Show work or explain your result. c) What is the x-intercept of this line? Show complete work.

106 106 LINEAR EQUATIONS AND FUNCTIONS Exercise 2 Writing Linear Equations from Graphs Use the given graph of the function f (x) to help answer the questions below. Assume that the line intersects grid corners at integer (not decimal) values. a) Is the line increasing, decreasing, or constant? b) What is the y-intercept? Plot and label the y- intercept on the graph. c) What is the x-intercept? Plot and label the x- intercept on the graph d) What is the slope? Show your work e) What is the equation of the line? Show complete work. Use function notation for your final answer.

107 3.4 WRITING THE EQUATION OF A LINE 107 Problem 2 : Class/Media - Applications of Linear Functions A candy company has a machine that produces candy canes. The number of candy candes, C, produced, depends on the amount of time, t, the machine has been operating. The machine produces 160 candy canes in five minutes. In twenty minutes, the machine can produce 640 candy canes. a) Determine a linear equation, in function notation, that models this situation. b) Determine the vertical intercept of this linear equation. Write it as an ordered pair and interpret its practical meaning in a complete sentence. c) Determine the horizontal intercept of this linear equation. Write it as an ordered pair and interpret its practical meaning in a complete sentence. d) How many candy canes will this machine produce in 1 minute? e) How many candy canes will this machine produce in 1 hour?

108 108 LINEAR EQUATIONS AND FUNCTIONS Exercise 3 Application of Linear Functions The graph below shows a person's distance from home as a function of time. a) Indentify the vertical intercept. Write it as an ordered pair and interpret its practical meaning in a complete sentence. b) Indentify the horizontal intercept. Write it as an ordered pair and interpret its practical meaning in a complete sentence. c) Let t represent time, and D represent the distance from home. Determine the linear equation that models this situation. Write your solution using function notation. d) How far has this person traveled in one minute?

109 3.4 WRITING THE EQUATION OF A LINE 109 Two lines are parallel if they never intersect. It is clear that if two lines are parallel, then their slopes must be equal. The picture below will help us understand the relatioship between the slopes of two lines that are perpendicular, that it, two lines that intersect at a rigght angle 90 o. The slope of the line L(x) is m 1 = b a. The slope of the line f (x) is m 2 = a b Note that the slope of line f (x) is negative since the line is decreasing. We can also see that the two right triangles are copies of each other, since they have all 3 sides of the same length, so the angles have the same measure as well. Notice that the sum of the angles so we have α +β+90 o =180 o, α +β=90 o Now let's take a close look at the angle wehre both lines intersect. The the angles α, β and γ must also add to 180 o. So we have α +β+γ = 180 o 90 +γ = 180 o γ = 90 o So the lines L(x) and f (x) are perpendicular. Slopes of Parallel and Perpendicular Lines: Slopes of parallel lines are equal to each other. That is, if one line has slope m 1 and another line has slope m 2 and both lines are parallel to each other, then m 1 =m 2. Slopes of perpendicular lines are negative reciprocals of each other. That is, if one line has slope m 1 and another line has slope m 2 and both lines are perpendicular to each other, then m 2 = 1 m 1.

110 110 LINEAR EQUATIONS AND FUNCTIONS Problem 3 : Class/Media - Writing Equations of Parallel and Perpendicular Lines For each of the following, find the equation of the line that meets the following criteria. Write your answer in function notation, if possible. a) Line contains the point (7, 3) and is parallel to y =2x +6 b) Line contains the point ( 4, 3) and is perpendicular to 2x +3y =6 c) Line contains the point (1,0) and is parallel to x =8

111 3.4 WRITING THE EQUATION OF A LINE 111 Exercise 4 Writing Equations of Parallel and Perpendicular Lines For each of the following, find the equation of the line that meets the following criteria. Write your answer in function notation, if possible. a) Line is parallel to 3x y = 7 and passes through the point (6, 5) b) Line is perpendicular to y + 3x = 2 and passes through the point ( 6,1) c) Line is parallel to y = 4 and passes through the point (2,7)

112 112 LINEAR EQUATIONS AND FUNCTIONS 3.5. SYSTEMS OF LINEAR EQUATIONS Problem 1 : Class/Media - How Many and What Kind of Solutioin? Below are graphs of 3 different systems of linear equations. Complete the chart below. Graph Number of Solutions Solution a) b) c)

113 3.5 SYSTEMS OF LINEAR EQUATIONS 113 Problem 2 : Class/Media - Solving Systems of Linear Equations Solve the following system of linear equations. Write your answer as an ordered pair. a) 3x =y+2 { 2x +y= 1 b) f (x) = 3x +2 { y =3x Exercise 1 Solving Systems of Linear Equations Solve the following system of linear equations. Write your answer as an ordered pair. a) b = 2a +10 { 4a+ 5b= 20 b) f (x) =4x +12 { 2x +y=6 c) 2m +3n = 1 { 4m =7 6n d) g(x)= 3x + 2 f (x)= 5 3 x

114 114 LINEAR EQUATIONS AND FUNCTIONS Systems of linear equations are often encountered in subject areas such as chemistry, physics, business, and more. Knowing how to solve these systems is very important. We will focus on 3 types of application problems: business applications, mixture problems and problems involving distance, rate and time. Remember that distance = rate time, that is, d =rt. Example 1 Application of Linear Equations Everyday, Mary walks from her home to the bus stop that is 5000 feet away. To get to her bus stop, she needs to walk on grass and pavement. She can walk at a rate of 5 ft/sec on pavement. On grass, she can only walk at a rate of 2.5 ft/sec since it is a favorite place for dogs. If she spends twice as long walking on grass than on pavement, find how much time (in minutes) was spent walking on each surface. (Round answers to the nearest tenth) Solution. Remember that Distance = Rate x Time that is, d =rt. It is always a good idea to set up a table. Let x = time spent walking on grass and y = time spent walking on pavement Surface Distance Rate Time Grass 2.5x 2.5 x Pavement 5y 5 y We know that Distance traveled on grass + distance traveled on pavement = 5000 feet. Therefore, 2.5x + 5y = We also know that she walked twice as long on grass than on pavement. Therefore, x =2y. We now have the system of linear equations: { 2.5x + 5y = 5000 x = 2y Using the substitution method, substitute x = 2y into the first equation. 2.5(2y) +5y = y +5y = y = 5000 y = 500 seconds Convert to minutes: y = 500 seconds x = 8.3 minutes 1 minute 60 seconds

115 3.5 SYSTEMS OF LINEAR EQUATIONS 115 Substitute y = 500 into the second equation: x = 2(500) x = 1000 seconds x = 1000 seconds x x = 16.7 minutes 1 minute 60 seconds Therefore, everyday, Mary walks 8.3 minutes on pavement and 16.7 minutes on grass to get to her bus stop. Problem 3 : Class/Media - Application of Systems of Linear Equations a) If 12% hydrochoric acid (HCl) is mixed with 40% HCl, how much of each is needed to make 100 liters that is 33% HCl? b) Two cars leave Seattle, one for Portland, traveling at an average speed of 62 mph and the other for Bellingham, going at an average speed of 68 mph. How long will it take before the two cars are 156 miles apart?

116 116 LINEAR EQUATIONS AND FUNCTIONS Exercise 2 Application of Systems of Linear Equations a) A lab has two solutions. One is made up of 10% sulfuric acid and the other, 30% sulfuric acid. How many milliliters (ml) of each is needed to make 200 ml that is 15% sulfuric acid solution? b) You are debating between two phone plans for your family of 4. Both plans include unlimited talk, text, 8 GB of data and a new smart phone for each member of your family. The first plan costs $95 a month with an upfront fee of $ The second plan costs $ a month with a $92.97 upfront fee. How many months will it take before the two plans cost the same? How much will you have spent by then?

117 PRACTICE PROBLEMS 117 PRACTICE PROBLEMS Section 3.1: Linear Equations and Functions 1. Find the slope of the line that passes through the given points. Then, determine if the line is increasing, decreasing or constant. Points Slope Slope Sign (+,, 0) Increasing, Decreasing or Constant (3, 2) and (6,8) m= = 6 3 =2 Positive Increasing a) ( 2,6) and ( 6, 2) b) (3, 5) and (7, 7) c) ( 1, 5) and (4,7) d) ( 3, 12) and (5, 1) e) ( 3 2,2 ) and ( 5 2,2 ) f) ( 3 4, 2 7) and ( 1 4, 4 7)

118 118 LINEAR EQUATIONS AND FUNCTIONS 2. Complete the table below. If the equation is not in slope-intercept form, show the steps required to convert it to that form. Also show work required to calculate the x-intercept. Write all intercepts as ordered pairs. Equation y = mx +b Form Slope y-intercept x-intercept a) y = 4x 8 b) x + y = 0 c) 3y + 6= x d) 5x y=2 e) 6x +3y =10 f) x = 1 2 y g) y = 4 h) x = 3

119 PRACTICE PROBLEMS 119 Section 3.2: Graphs of Linear Functions 3. For the given linear functions, complete the table of values. Plot the ordered pairs and graph the line. a) y = 3x 2 x y =3x 2 Ordered Pair 3 y = 3( 3) 2= 11 ( 3, 11) b) y = 2x + 4 x y = 2x + 4 Ordered Pair

120 120 LINEAR EQUATIONS AND FUNCTIONS c) y = 3 2 x +1 x y = 3 x + 1 Ordered Pair d) y = 2 5 x 3 x y = 2 5 x 3 Ordered Pair

121 PRACTICE PROBLEMS Use the intercepts to draw the graph of each equation. Show your work to find these points. Plot and label the intercepts on the graph and then draw a line through the points. a) y = 3x 1 y-intercept: (, ) x-intercept: (, ) b) y = x + 2 y-intercept: (, ) x-intercept: (, )

122 122 LINEAR EQUATIONS AND FUNCTIONS c) y = 1 3 x +3 y-intercept: (, ) x-intercept: (, ) d) 2x 3y = 12 y-intercept: (, ) x-intercept: (, )

123 PRACTICE PROBLEMS The slope and a point on the line is given. Use the information to draw the graph of the line. Identify at least two additional points on the line and label them on the graph. Use the information to draw the graph of the line. Finally, complete the table shown. a) Line has a slope of 4 and contains the point ( 3,0) - Use the slope to find 2 additional points to graph the line. Label them on the graph. - Draw the line through the points you found. -Use the slope to complete the table x y b) Line has a slope of 2 and contains the point ( 1,5) - Use the slope to find 2 additional points to graph the line. Label them on the graph. - Draw the line through the points you found. -Use the slope to complete the table x y

124 124 LINEAR EQUATIONS AND FUNCTIONS c) Line has a slope of 1 and contains the point (2, 4) 3 - Use the slope to find 2 additional points to graph the line. Label them on the graph. - Draw the line through the points you found. -Use the slope to complete the table x y 2 4 d) Line has a slope of 1 and contains the point ( 3, 2) 2 - Use the slope to find 2 additional points to graph the line. Label them on the graph. - Draw the line through the points you found. -Use the slope to complete the table x y

125 PRACTICE PROBLEMS 125 Section 3.3: Horizontal and Vertical Lines 6. Complete the table. Equation Slope x-intercept y-intercept y = 5 m= 0 Does not exist (0,5) a) y = 3 b) x =3 c) y = 2 d) x = 4 e) y = 0 f) x =0

126 126 LINEAR EQUATIONS AND FUNCTIONS 7. Graph each of the following. Plot and label any intercepts. a) y = 3 b) x = 3 c) y = 2 d) x = 4 e) y = 0 f) x = 0

127 PRACTICE PROBLEMS Use the given information to determine the equation of the line and then graph the line. a) Given Equation Graph Horizontal line that passes through the point ( 2, 3) Vertical line that passes through the point (5, 2) b) c) Line with zero slope passing through the point ( 2, 4) d) Line with undefined slope passing through the point ( 1, 4)

128 128 LINEAR EQUATIONS AND FUNCTIONS Section 3.4: Writing Equation of a Line 9. Find the equation of the line that meets the following criteria. Slope Point Equation of Line a) m= 2 (0, 3) b) m = 4 ( 0, 2 3) c) m= 3 8 (0, 5) d) m = 2.37 (0, 6.35) 10. Find the equation of the line that meets the following criteria. Slope Point Equation of Line Function Notation a) m =2 (2, 3) b) m= 4 (3, 4) c) m = 5 16 ( 8, 5) d) m = 1.4 (2, 2.34)

129 PRACTICE PROBLEMS Find the equation of the line that meets the following criteria. Two Points Slope Equation of Line Function Notation a) (2, 3) and (4, 7) b) ( 3, 6) and (3, 12) c) (5, 5) and ( 1,3) d) (2, 4.2) and (6, 9.4) 12. Determine the equation of the line that is parallel to the given line and passes through the given point. Equation of Given Line Point on Parallel Line Slope of Parallel Line Equation of Parallel Line Function Notation a) y = 3 2 x + 2 (2, 3) b) y = 3x + 4 (3, 4) c) y =3 ( 6, 5)

130 130 LINEAR EQUATIONS AND FUNCTIONS 13. Determine the equation of the line that is perpendicular to the given line and passes through the given point. Equation of Given Line Point on Perpendicular Line Slope of Perpendicular Line Equation of Perpendicular Line Function Notation a) y =2x 4 (6, 3) b)y = 1 3 x +1 ( 3, 4) c) x = 3 ( 1, 2)

131 PRACTICE PROBLEMS Determine the equation of the line that corresponds to the given graph. Graph Equation a) b) c)

132 132 LINEAR EQUATIONS AND FUNCTIONS 15. Use the given graph to help answer the questions below. a) Is the line above increasing, decreasing or constant? b) What is the y-intercept? Plot and label the point on the graph. c) What is the x-intercept? Plot and label the point on the graph. d) What is the slope? e) What is the equation of the line in y = mx + b form?

133 PRACTICE PROBLEMS 133 Systems of Linear Equations 16. Solve the following systems of linear equations. Write your answer as an ordered pair. x 3y = 9 a) { f (x)= 1 2m 7n = 4 3 x b) { m 2n =1 c) g(x)= x 5 { 2x y = 5 d) a =3b +5 { 3a 9b = Sammy owns a small crepe shop. His monthly rent is $1410 and each crepe costs him $6.50. Sammy sells each crepe for $ How many crepes does Sammy have to sell each month in order to break even?

134 134 LINEAR EQUATIONS AND FUNCTIONS 18. How much of the $0.87 per pound bird seed needs to be mixed with the $0.62 per pound bird seed to get a 200 lb. mixture that can sell for $0.75 per pound? 19. A husband and wife are on the Burke Gilman Trail on a beautiful Saturday. Unfortunately, the wife's bicycle has a flat so the husband jogged ahead at an average speed of 6 mph. Half an hour later, the wife starts biking at an average speed of 10 mph. How long will it take for the wife to catch up with her husband? 20. How many liters of a 15% alcohol solution must be mixed with 30 liters of a 23% alcohol solution to get a 20% alcohol solution?

135 PRACTICE PROBLEMS Twice a week, Mark jogs for 30 minutes and then walks for 90 minutes. He usually jogs at a speed that is 8 mph faster than his walking speed and covers a distance of 10 miles. Find how far he walks and how far he jogs. 22. You buy coffee beans at wholesale prices, blend them and resell them to retail stores. Arabica coffee beans can be bought at $1.72 per pound and robusta coffee beans at $0.97 per pound. The beans are to be mixed to get 500 pounds of a blend that will sell for $1.34 per pound. How many pounds of each should be used? 23. Joey is comparing cellphone plans that offer unlimited talk, text and 1 GB of data. Company A costs $51/month with an upfront fee of $18. Company B has no upfront fee but costs $60/month. How many months will it take before you pay the same price for either plan?

136 136 LINEAR EQUATIONS AND FUNCTIONS 24. The hare and the tortoise story is about the hare bragging about how fast he can run. Tired of hearing the hare boast, the tortoise challenges the hare to a race. The hare thinks he has plenty of time, so he sleeps while the tortoise starts towards the finish line at a speed of 0.17 mph. Suppose the hare wakes up after a very long 6 hours of nap and starts sprinting towards the finish line at a speed of 35 mph. a) How long does it take for the hare to catch up with the tortoise? Give your answer in minutes, rounded to the nearest hundredth. b) How far did the tortoise go when the hare overtook him? Give answer in miles, rounded to the nearest hundredth. c) If the race is 1 mile long, did the tortoise win? 25. Cora decided to go on a diet. On the day she started, she weighed 200 pounds. For the next 8 weeks, she consistently lost 2 pounds a week. At the end of 8 weeks, she decided to make a graph showing her progress. Let W(n) be Cora's weight in pounds after n weeks of dieting. Find the equation, in function notation, of the line for the above problem. Graph the result.

137 PRACTICE PROBLEMS 137 Chapter 3 - Extension 26. Mark needed 200 pounds of roofing nails for his project. He poured one cup filled with nails into a bucket and found that it weight 2.3 pounds. He then poured 4 more cups of nails into the bucket and found that it weighed 9.5 pounds. He figured if he used the points (1, 2.3) and (5, 9.5), he could figure out a formula (i.e., equation) and calculate how many cups he would need. a) Find the equation, in function notation, of the line for this problem. Let W(c) be the combined weight of the bucket and c cups of nails. b) What is W(0)? Interpret the result in the context of the problem. c) How many cups of roofing nails does Mark need for his project?

138 138 LINEAR EQUATIONS AND FUNCTIONS CHAPTER 3 ASSESSMENT 1. Determine the equation of the line through the points (4,3) and (12, 3). Write your answer in slope intercept form, f (x) = mx + b. 2. The function P(n) = 455n 1820 represents a computer manufacturer's profit when n computers are sold. a) Identify the vertical intercept. Write it as an ordered pair. Then in a complete sentence, interpret its practical meaning in the context of the situation. b) Determine the horizontal intercept. Write it as an ordered pair. Then in a complete sentence, interpret its practical meaning in the context of the situation. 3. Determine the equation of the vertical line passing through the point (4,7). 4. The x-axis is a line. Write the equation of this line.

139 CHAPTER 3 ASSESSMENT Fill in the table below. Intercepts must be written as ordered pairs. For behavior, I = Increasing, D = Decreasing, C = Constant (Horizontal), V = Vertical Equation Slope Vertical Intercept Horizontal Intercept Behavior (I, D, C, V) a) y = 2x 16 b) f (x) =8 3x c) y = 2 d) 1 (0, 0) e) (0, 9) ( 3, 0) f) undefined (6, 0) g) 2 3 ( 12, 0) h) (0, 4) (2,0) 6. Solve the following systems of linear equations algebraically and write the solution as an ordered pair, if possible. Then verify the solution by graphing each line. a) f (x)= 2 x 3 { g(x)= 14 4x b) { 2y x = 3 2x 4y = 9

140

141 CHAPTER 4 LINEAR FUNCTIONS AND APPLICATIONS In this chapter, we take a closer look at linear functions and how real world situations can be modeled using linear functions. We study the relationship between average rate of change and slope and how to interpret these characteristics. We also learn how to create linear models from data sets using linear modeling. Chapter Topics: Section Review of Linear Functions Section Average Rate of Change Average Rate of Change as Slope Interpret Average Rate of Change Use Average Rate of Change to Determine if Function is Linear Section Determining Equation of a Line Point-Slope Form Section Scatterplot and Linear Modeling Drawing Scatterplots Determining Equation of Best Fit Line Using Line of Best Fit to Solve Application Problems 141

142 142 LINEAR FUNCTIONS AND APPLICATIONS Component Required? Yes or No Comments Due Score Mini-Lesson Homework Quiz Test Practice Problems Chapter Assessment

143 4.1 REVIEW OF LINEAR FUNCTIONS REVIEW OF LINEAR FUNCTIONS This lesson will combine the concepts of functions and linear equations. To write a linear equation as a linear function, replace the dependent variable using the function notation. For example, in the following linear equation, y = mx + b, we replace the variable y with f (x). So y = mx + b written as a linear function becomes f (x) = mx + b. Important Things to Remember about the Linear Function f (x) =mx +b x represents the input quantity f(x) represents the output quantity The graph of f (x) is a straight line with slope, m, and y-intercept (0, b) Given any two points (x 1, y 1 ) and (x 2,y 2 ) on a line, m = Change in Output Change in Input = y2 y1 Change in y = x 2 x 1 Change in x = y x If m >0, then the graph increases from left to right. If m <0, then the graph decreases from left to right. If m =0, then f (x) is a constant function and the graph is a horizontal line. The domain of a linear function is generally all real numbers unless a context or situation is applied in which case we interpret the practical domain in that context or situation. One way to identify the y-intercept is to evaluate f (0). In other words, substitute 0 for the input, x, and determine the resulting output. Then (0,y) is the y-intercept. To find the x-intercept, solve the equation f (x) = 0 for x. In other words, set mx + b = 0 and solve for the value of x. Then (x,0) is the x-intercept.

144 144 LINEAR FUNCTIONS AND APPLICATIONS Exercise 1 Review of Linear Functions The function E(t)= t gives the surface elevation (in feet above sea level) of Lake Powell t years after a) Identify the vertical intercept of this linear function and write a sentence explaining its meaning in this situation. b) Determine the surface elevation of Lake Powell in the year Show your work and write your answer in a complete sentence. c) Determine E(4) and write a sentence explaining the meaning of your answer. d) Is the surface elevation of Lake Powell increasing or decreasing? How do you know? e) This function accurately models the surface elevation of Lake Powell from 1999 to Determine the practical range of this linear function.

145 4.2 AVERAGE RATE OF CHANGE AVERAGE RATE OF CHANGE Average Rate of Change The average rate of change of a function over a specified interval is the ratio: Average Rate of Change = Change in Output Change in Input = Change in y Change in x = y x Units for the Average Rate of Change are always output units per input units. Output Units, which can be interpreted as Input Units Problem 1 : Class/Media - Average Rate of Change The function E(t) = t gives the surface elevation of Lake Powell t years after Complete the table below. t (years since 1999) 0 E(t) - Surface Elevation of Lake Powell (in feet above sea level) a) Determine the average rate of change of the surface elevation between 1999 and 2000.

146 146 LINEAR FUNCTIONS AND APPLICATIONS b) Determine the average rate of change of the surface elevation between 2000 and c) Determine the average rate of change of the surface elevation between 2001 and d) What do you notice about the average rates of change for the function, E(t)? e) On the grid below, draw a GOOD graph of E(t) with all appropriate labels. Note. Because the average rate of change is constant for these depreciation data, we say that a LINEAR FUNCTION models these data best.

147 4.2 AVERAGE RATE OF CHANGE 147 Does the average rate of change look familiar? It should! Another word for average rate of change is slope. Given any two points, (x 1, y 1 ) and (x 2, y 2 ) on a line, the slope is determined by Change in Output y2 y1 computing the following ratio: m = = Change in Input x 2 x 1 Therefore, AVERAGE RATE OF CHANGE = SLOPE over a given interval. Average Rate of Change Given any two points, (x 1,y 1 ) and (x 2,y 2 ) on a line, the average rate of change between the points on the interval x 1 to x 2 is determined by computing the following ratio: Change in Output Change in y Average Rate of Change = = Change in Input Change in x = y x If the function is LINEAR, then the average rate of change will be the same between any pair of points. If the function is LINEAR, then the average rate of change is the SLOPE of the linear function. Problem 2 : Class/Media - Is the Function Linear? For each of the following, determine if the function is linear. If it is linear, give the slope. a) x y b) x y c) x y

148 148 LINEAR FUNCTIONS AND APPLICATIONS Exercise 1 Is the Function Linear? For each of the following, determine if the function is linear. If it is linear, give the slope. a) x f(x) b) x A(x) c) x g(x)

149 4.3 DETERMINING THE EQUATION OF A LINE DETERMINING THE EQUATION OF A LINE Equation of a Line in Point-Slope Form The point-slope form of a line is: y y 1 = m(x x 1 ) where m is the slope of the line and (x 1, y 1 ) is a point on the line. Example 1 Determining the Equation of a Line Determine if the data below represent a linear function. If so, determine the equation that best fits the given data. Solution. x y Compute a few slopes to determine if the data are linear: Between (1, 75) and (5, 275), m= 5 1 = = Between (5, 275) and (9, 475), m= = = Between (9, 475) and (13, 675), m= 13 9 = = 50 The data appear to be linear with a slope of 50. Determine the Linear Equation Using Point-Slope Form: Once you have the slope, m, of the line, you can use the point-slope form to find the linear equation. The point-slope form is y y 1 =m(x x 1 ). In our example, m=50 and we can use any of the points. Let us use the point (1, 75). This gives us: y 75 = 50(x 1) y 75 = 50x 50 y = 50x + 25 In Function Notation: f (x) = 50x + 25

150 150 LINEAR FUNCTIONS AND APPLICATIONS Problem 1 : Class/Media - Average Rate of Change and Linear Functions The data below represent your annual salary for the first four years of your current job. Time, t (in years) Salary, S (in $1000) a) Identify the vertical intercept. Write it as an ordered pair and interpret its meaning in a complete sentence. b) Determine the average rate of change during this 4-year time period. Write a sentence explaining the meaning of the average rate of change in this situation. Be sure to include units in your answer. c) Verify that the data represent a linear function by computing the average rate of change between two additional pair of points. d) Write the linear function that models the data. Use the indicated variables and proper function notation.

151 4.3 DETERMINING THE EQUATION OF A LINE 151 Exercise 1 Average Rate of Change The data below show a person's body weight during a 5-week diet program. Time, t (in weeks) Weight, W (in pounds) a) Identify the vertical intercept. Write it as an ordered pair and write a sentence explaining its meaning in this situation. b) Compute the average rate of change for the 5-week period. Be sure to include units. c) Write a sentence explaining the meaning of your answer in part (b) in the given situation. d) Do the data points in the table define a perfectly linear function? Why or why not? e) On the grid below, draw a good graph of this data set with all appropriate labels.

152 152 LINEAR FUNCTIONS AND APPLICATIONS 4.4. SCATTERPLOTS AND LINEAR MODELING Just because data are not perfectly linear does not mean we cannot write an approximate linear model for the given data set. In fact, most data in the real world are not perfectly linear and all we can do is write models that are close to the given values. If you take a statistics class, you will learn a lot more about this process. In this class, you will be introduced to the basics. This process is also called finding the line of best fit. Exercise 1 The Line of Best Fit Below are the scatterplots of different sets of data. Notice that not all of them are perfectly linear but the data seem to follow a linear pattern. Using a ruler or straightedge, draw a straight line on each of the graphs that appear to fit the data best. (Note that this line might not actually touch all of the data points.) The first one has been done for you.

153 4.4 SCATTERPLOTS AND LINEAR MODELING 153 a) b) c) d)

154 154 LINEAR FUNCTIONS AND APPLICATIONS Linear Modeling To determine a linear equation that models the given data, we could do a variety of things. In this class, we will find the equation of the line that goes through at least two good points. That is, our line will be as close to as many data points as possible and will go through at least 2 given data points. The line that goes through these points will be a good line to fit the data. There may be many options for which to choose the two points. However, there are also pairs of points that would give us a bad line. We have to choose wisely. Criteria for a GOOD Graph: 1. Horizontal axis should be properly labeled with the name and units of the input quantity. 2. Vertical axis should be properly labeled with the name and units of the output quantity. 3. Appropriate scale should be used. Start at or just below the lowest value. End at or just above the highest value. Scale the graph so the adjacent tick marks are of equal distance. Use numbers that make sense for the given data set. Axes must meet at (0, 0). Use a // or a sawtooth mark between the origin and the first tick mark to denote a jump in the range of values. 4. All points should be plotted correctly. 5. Graph should be neat and uncluttered. Use a ruler or straight-edge to draw the best fit line.

155 4.4 SCATTERPLOTS AND LINEAR MODELING 155 Problem 1 : Class/Media - Linear Modeling Consider the data set from a person following a 5-week diet program. Time, t (in weeks) Weight, W (in pounds) a) Draw a scatterplot of the data set below. You will have to first decide a scale for each axis. Then draw a line that you think will be a good line to fit these data. b) Find a linear function that models the situation. Use the indicated variables and proper function notation. c) Use your linear model to predict the weight of the person if he continued the program for one more week.

156 156 LINEAR FUNCTIONS AND APPLICATIONS Exercise 2 Linear Modeling The function f (n) is defined by the following table. n f(n) a) Based on this table, determine f (6). Write your answer as an ordered pair. b) Draw a scatterplot of the data below. You will have to first decide a scale for each axis. Then draw a line that you think will be a good line to fit these data. c) Find the equation of the best fit line for the given data. Round to two decimal places as needed. Use the indicated variables and proper function notation. d) Using your line of best fit, determine f (6). Write the specific ordered pair associated with this result. e) Your answers for (a) and (e) may be different. Why is this the case?

157 4.4 SCATTERPLOTS AND LINEAR MODELING 157 Exercise 3 Linear Modeling The following table gives the total number of live Christmas trees sold, in millions of trees, in the United States from 2005 to Find a line of best fit. (Source: Statista.com) Year Christmas Trees Sold Let t represent the number of years since 2005 and C(t) be the total number of Christmas trees sold in the US, in millions of trees. a) Start by entering new t values for the table below based upon the number of years since The first few are done for you. t 0 1 C(t) b) Draw a scatterplot of the data below. You will have to first decide a scale for each axis. Then draw a line that you think will be a good line to fit these data.

158 158 LINEAR FUNCTIONS AND APPLICATIONS c) Find the equation of the best fit line for the given data. Round to one decimal place. Use the indicated variables and proper function notation. d) Use the equation to determine C(6) and in a complete sentence, explain its meaning in the context of this situation. e) Use the equation to predict the number of Christmas trees that will be sold in the year Write your answer in a complete sentence. f) Identify the slope of the equation and in a complete sentence, explain its meaning in the context of this situation.

159 PRACTICE PROBLEMS 159 PRACTICE PROBLEMS Section 4.1: Review of Linear Function 1. Edward the vampire can run at a speed of 70 miles per hour. His girlfriend, Bella, is 875 miles away from Edward, visiting her mom in Phoenix. Edward devides to visit her. Edward's distance, D, from Bella, t hours after he leaves for his trip can be modeled by the linear function D(t) = 70t a) Find the vertical intercept of the function. Interpret its meaning in the context of the problem. b) Find the horizontal intercept of the function. Interpret its meaning in the context of the problem. c) Evaluate D(4). Interpret its meaning in the context of the problem. d) Find the value of t for which D(t) =504. Interpret its meaning in the context of the problem. e) Is the function, D, increasing or decreasing? How do you know? f) Determine the slope or rate of change of the function D. (Be sure to include the units). What does the slope or rate of change represent in the context of the problem? g) Determine the practical domain and practical range of this function. Assume that t 0 and that Edward stops traveling when he reaches Bella in Phoenix. Practical Domain: Practical Range: t D(t)

160 160 LINEAR FUNCTIONS AND APPLICATIONS Section 4.2: Average Rate of Change 2. For each of the following functions, determine if the function is linear or not. If it is linear, give the slope. a) x f(x) b) x g(x) c) t s(t) d) x h(x) e) n p(n)

161 PRACTICE PROBLEMS 161 Section 4.3: Determining Equation of a Line 3. The data below represent the number of times your friend's embarrassing video has been viewed per hour since you uploaded it. The data are exactly linear. Time, t (in hours) Views, V , , , 800 a) Identify the vertical intercept and average rate of change for the data. b) Use the results from part (a) to write the linear function that represents the data. Use the indicated variables and proper function notation. c) Use your function to determine the number of views in hour 8. Write your final result in a complete sentence. d) Use your function to determine how many hours until the number of views reaches 100,000. (Round to the nearest whole hour). Write your final result in a complete sentence.

162 162 LINEAR FUNCTIONS AND APPLICATIONS 4. You adopted an adult cat four years ago and have been lax on weight management. The data below represent your cat's weight for four years she's lived with you. The data are exactly linear. Time, t (in years) Weight, W (in pounds) a) Identify the vertical intercept and average rate of change for the data. b) Use your results from part (a) to write a linear function that represents the data. Use the indicated variables and proper function notation. c) Use your function to determine how much the cat will weigh in year 8. Write your final result in a complete sentence. d) Use your function to determine how many years it would take for your cat to reach 20 pounds. Round to the nearest whole year.

163 PRACTICE PROBLEMS The data below represent how many push-up Tim can do in a minute at the start of a 5-week exercise program and each week thereafter. Time, t (in weeks) Push-Ups per minute, P a) Compute the average rate of change for weeks 0 and 3. (Include the units in your answer) b) Compute the average rate of change for weeks 1 and 4. (Include the units in your answer) c) Compute the average rate of change for weeks 0 and 5. (Include the units in your answer) d) What is the meaning of the average rate of change in this situation? Answer in a complete sentence. e) Do the data points in the table define a perfectly linear function? Why or why not?

164 164 LINEAR FUNCTIONS AND APPLICATIONS 6. You decide to save up for a vacation to Europe by throwing all your loose change in a large coffee can. After a few months, you discover that the jar is 2 inches full and contains $124. a) Determine the average rate of change, in $ (dollars per inch), for the coffee can inch from when it was empty (0 inches) to when it was 2 inches deep. b) A month later, you check the can and find the change is 3 inches deep and contains $186. Find the average rate of change, in $, for the coffee can from 0 inches to inch 3 inches. c) What is the meaning of the average rate of change in this situation? You do some additional calculation and create a table for the can of change. d, depth of the change (in inches) V, Money in the can (in $) d) Use the information found so far to write an equation that describes this situation. Use the indicated variables and the proper function notation. e) You need $1000 for your vacation. In a complete sentence, state how deep the change has to be to reach your goal. Also write the result as an ordered pair and in function notation.

165 PRACTICE PROBLEMS 165 Section 4.4: Scatterplot and Linear Modeling 7. Create a scatterplot of the data set. If the data appear to be linear, draw the best fit line. x y Create a scatterplot of the data set. If the data appear to be linear, draw the best fit line. x y

166 166 LINEAR FUNCTIONS AND APPLICATIONS 9. Sara is selling cookies to raise funds for her school orchestra. They cost $4 per box. The table below shows how much Sara has earned, E, based on the number of days, t, she has been selling cookies t E(t) a) Find the average rate of change for the following pairs of t values. i. t = 0 and t = 2 ii. t = 2 and t = 7 iii. t = 3 and t = 12 b) Based on your answers to part (a), is it possible that the data are exactly linear? Explain. c) Create a scatterplot for the data.

167 PRACTICE PROBLEMS 167 d) Based on your scatterplot in (c), do the data appear to be linear? Explain. e) Find a linear function that models the situation. Use the indicated variables and proper function notation. f) Explain the meaning of the slope of your linear equation in a complete sentence. How does it compare to the average rate you found in part (a)?

168 168 LINEAR FUNCTIONS AND APPLICATIONS 10. Jose is recording the average daily temperature for his science class during the month of June in Phoenix, Arizona. The table below represents the average daily temperature t days after June 1. t D(t) a) Find the average rate of change for the following pairs of t values. i. t = 0 and t = 1 ii. t = 1 and t = 5 iii. t = 3 and t = 10 b) Based on your answers to part (a), do the data appear linear? Explain. c) Create a scatterplot for the data. d) Based on your scatterplot in (c), do the data appear to be linear? Explain.

169 PRACTICE PROBLEMS Tamara is collecting donations for her local food bank. The data below represents the pounds of food, P, in the food bank t days after November 1. t P(t) a) Find the average rate of change for the following pairs of t values. i. t = 0 and t = 1 ii. t = 1 and t = 6 iii. t = 3 and t = 15 b) Based on your answers to part (a), do the data appear linear? Explain. c) Create a scatterplot for the data. d) Based on your scatterplot in (c), do the data appear to be linear? Explain.

170 170 LINEAR FUNCTIONS AND APPLICATIONS 12. The following table shows the number of newspaper subscription in Middletown, USA where t represents the number of years since 2002 (where t =0 in 2002) and S(t) represents the total subscriptions each year, measured in thousands. t S(t) a) Draw a scatterplot for the data. You will have to first decide a scale for each axis. b) Based on your scatterplot, do the data appear to be exactly linear, approximately linear or not linear? Explain. c) If the data appear linear, draw a line you think would be a good line to best fit these data. d) Find a linear function that models the situation. Use the indicated variables and proper function notation. (Round to 2 decimal places, as needed.

171 PRACTICE PROBLEMS 171 e) What is the slope of your linear model? Interpret its meaning in the context of the problem. f) What is the vertical intercept? Interpret its meaning in the context of the problem. g) Use your linear equation to estimate the total number of subscription in Show your computation and your final result. h) Use your linear equation to estimate the total number of subscription in How does this value compare to the data value in the table? Write in complete sentence. i) Use your linear equation to estimate the year in which the newspaper subscription will be 100,000. Round to the nearest whole year. (Remember S(t) is measured in thousands so you must solve S(t)= 100.)

172 172 LINEAR FUNCTIONS AND APPLICATIONS 13. Scott is hiking the Appalachian Trail from Georgia to Maine. The distance of his hike is 2200 miles. It took Scott 123 days to complete the hike. The data below represent the distance, D, he had hiked t days after the start of his trip. t (days hiking) D(t) (Distance in miles) a) Draw a scatterplot for the data. You will have to first decide a scale for each axis. b) Based on your scatterplot, do the data appear to be exactly linear, approximately linear or not linear? Explain. c) If the data appear to be linear, draw a line that you think would be a good line to fit the data. d) Find a linear function that models the situation. Use the indicated variables and proper function notation. (Round to 2 decimal places, as needed.

173 PRACTICE PROBLEMS 173 e) What is the slope of your model. Interpret its meaning in the context of this problem. f) Use your linear equation to estimate the total number of miles Scott has hiked in 50 days. g) Use your linear equation to estimate when Scott has hiked 1000 miles.

174 174 LINEAR FUNCTIONS AND APPLICATIONS Chapter 4: Extension 14. The table below shows the educational attainment of women in the labor force from 1970 to Year Percent of High School Graduates (No College) Percent of College Graduates , Let t represent the number of years since 1970, H(t) represent the percent of High School Graduates (No College) and C(t) represent the percent of College Graduates. a) Start by entering new t values for the table below based upon the number of years since Year H(t) C(t) , b) Draw a scatterplot of the above data. Use 2 different colored pens to draw the best fit line. c) Find a model for the percent of High School Graduates (No College). Use the indicated variables and proper function notation. d) What is the slope for this model (High School Graduates)? Interpret its meaning in the context of the problem. e) Using the linear equation obtained, find H(25). Interpret the result in the context of the problem. Check the reasonableness of your answer with your graph. f) Find a model for the percent of College Graduates. Use the indicated variables and proper function notation. g) What is the vertical intercept for this model (College Graduates)? Interpret its meaning in the context of the problem. h) Approximately what year were there 25% women college graduates in the labor force. First solve algebraically using the equation you obtaines. Then check the reasonableness of your answer with your graph.

175 PRACTICE PROBLEMS 175 i) Find C(53) algebraically. Interpret the result in the context of the problem. Then check the reasonableness of your answer with your graph. j) In what year is the percent of women High School Graduates equal to the percent of women College Graduates in the labor force? Solve algebraically using the linear equation you obtained. Then check the reasonableness of your answer with your graph. k) Approximately what year will the percent of women College Graduates surpass the percent of women High School Graduates in the labor force? 15. Your turn. Create a story problem where the data change linearly and then create a table that has data points for that story. a) Write the story. b) Create a table for the story problem. Make sure you use function notation. c) Compute the average rate of change for your data. Be sure to include units. d) What is the meaning of the average rate of change in this situation? e) Determine the vertical intercept for your data. Interpret its meaning in the context of the problem. f) Use the vertical intercept and the rate of change to write the linear function model for the data. Use proper variable names and proper function notation. g) Write a read the data question given the input. Write your question in a complete sentence and in function notation. h) Write a read the data question given the output. Write your question in a complete sentence and in function notation. i) Write a read between the data (Interpolating the Data) question given the input. Write your question in a complete sentence and in function notation. j) Write a read between the data (Interpolating the Data) question given the output. Write your question in a complete sentence and in function notation. k) Write a read beyond the data (Extrapolating the Data) question given the input. Write your question in a complete sentence and in function notation. l) Write a read beyond the data (Extrapolating the Data) question given the output. Write your question in a complete sentence and in function notation.

176 176 LINEAR FUNCTIONS AND APPLICATIONS CHAPTER 4 ASSESSMENT The total sales, S, in millions of dollars of a particular company since 1990 is given by the table below. Total Sales Year Since 1990 (in millions of dollars) Determine the sales in Write your answer in a complete sentence. 2. Plot the data and then draw the line that best fits the data. 3. Let S(t) represent the total sales of this company t years after Find the linear equation that models this situation and write it in function notation. (Round to 2 decimal places, when needed)

177 CHAPTER 4 ASSESSMENT Use your linear model to determine sales in Round your answer to the nearest hundredth. Write your answer in a complete sentence. 5. Your answers to question (1) and (4) may be different. Why is this the case? Answer in a complete sentence. 6. Use your linear model to determine the year in which sales would reach $3,000,000. Write your answer in a complete sentence. 7. Find S(30). In a complete sentence, interpret the meaning your result in the context of the situation.

178

179 CHAPTER 5 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES In this chapter, you will learn how to solve absolute value equations and inequalities. For inequalities, you will also learn how to write the solution in both inequality and interval notation, and graph the solution on a number line. You will be introduced to applications using absolute value. Pay special attention to the problems you are working on, in particular, note details such as signs and inequality symbols. Having a neat and organized work will pay off in this chapter. Chapter Topics: Section Absolute Value Equations Definition of Absolute Value Solving Absolute Value Equations Section Absolute Value Inequality Solving Absolute Value Inequality Writing Solutions Sets Graphing Solutions on a Number Line Section Absolute Value Applications 179

180 180 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES Component Required? Yes or No Comments Due Score Mini-Lesson Homework Quiz Test Practice Problems Chapter Assessment

181 5.1 ABSOLUTE VALUE EQUATIONS ABSOLUTE VALUE EQUATIONS Definition of Absolute Value: x = x, if x 0 { x, if x <0, for any real number x Let us take a look at 3 and 3 using the definition. 3 =3. Since 3 >0, according to the definition, we use the value 3. 3 = ( 3)= 3. Since 3 <0, according to the definition, we take the opposite of 3. Graph of Absolute Value Function: The graph of the absolute value function, y = x is made up of two parts. The first part is y = x, which is the graph of a line with slope 1 and y-intercept (0, 0). However, we only graph for values of x that are non-negative. The second part is y = x, which is the graph of a line with slope 1 and y-intercept (0, 0). However, we only graph for values of x that are negative.

182 182 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES Suppose we want to solve x =a for x. Let us graph y = x and y=a. Then look at the intersection of the two graphs. Three possible cases will occur. CASE 1: a is positive will result in two unique solutions Horizontal line, y = a, intersects the absolute value graph in 2 unique places: x = a and x =a When x =a, for a >0 then: x = a or x = a CASE 2: a equals zero will result in one unique solution Horizontal line, y = 0, intersects the absolute value graph in 1 unique place: x =0 When x =0, then x =0 CASE 3: a is negative will result in NO solution Horizontal line, y = a, does not intersect the absolute value graph. When x =a, for a< 0, then there is no Solution

183 5.1 ABSOLUTE VALUE EQUATIONS 183 Solution to x =a: CASE 1: When a is positive, then x = a or x =a. CASE 2: When a is zero, then x =0. CASE 3: When a is negative, then x has NO solution. To compute distance from 7 to 5, take the right-hand number and subtract the left-hand number. That is, 5 ( 7)= 12. Another way to solve absolute value equation is to visualize it using the idea of distance. CASE 1: a is positive Example Algebraic Solution Distance Idea x =3 We have 2 solutions: x =3 or x = 3 x = 3 x =3 is the same as x 0 =3 Go a distance of 3 units from 0 x 5 =6 We have 2 solutions: (x 5) = 6 or x 5 = 6 x 5 = 6 x = 11 x = 1 x 5 =6 means Go a distance of 6 units from 5

184 184 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES CASE 2: a is zero Example Algebraic Solution Distance Idea x =0 We have 1 solution: x =0 x =0is thesame as x 0 =0 Go a distance of 0 unit from 0 x +1 =0 We have 1 solution: x +1=0 x = 1 x + 1 =0 same as x ( 1) =0 Go a distance of 0 unit from 1 CASE 3: a is negative Example Algebraic Solution Distance Idea x = 8 x 8 Therefore, there is no solution x = 8 is the same as x 0 = 8 Go a distance of 8 units from 0 but we cannot go a negative distance. x 4 +7=2 Always isolate the absolute value equation first. In this case, subtract 7 from each side of the equation first. We get x 4 = 5 but x 4 5. Therefore, there is no solution to this problem. x 4 = 5 means go a distance of 5 units from 4 but we cannot go a negative distance. Problem 1 : Solving Absolute Value Equations Solve the following equations. 1 3 a) x + = 2 4 b) 3x +1=7

185 5.1 ABSOLUTE VALUE EQUATIONS 185 Exercise 1 Solving Absolute Value Equations Solve the following equations. a) x 7 =8 b) x + 1 = 3 c) 2x 5 =7 d) x = 3 e) 5 x + 2 x 6= 2 f) = 4 3 3

186 186 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES 5.2. ABSOLUTE VALUE INEQUALITIES When solving x a(or x <a) for x, there are 2 possible cases. CASE 1: a is positive x <a Inequality Notation: a x a Interval Notation: [ a, a] Number Line: CASE 2: a is negative x <a The Inequality has no solution. The absolute value graph is never less than y = a. Absolute value graph is always above y =a. The solution to x a (or x < a): CASE 1: When a is positive, the solution is as follows: Inequality Notation: a x a (or a< x < a). Interval Notaion: [ a, a] (or ( a, a)) CASE 2: When a is negative, then the inequality has no solution.

187 5.2 ABSOLUTE VALUE INEQUALITIES 187 Another way to solve absolute value inequality is to visualize it using the idea of distance. CASE 1: a is positive Example Algebraic Solution Distance Idea x 4 Inequality Notation: 4 x 4 Interval Notation: [ 4, 4] x 4 is the same as x 0 4. That is, go a distance of at most 4 units from 0 x 5 <6 Solution: 6< x 5< 6 1< x < 11 Inequality Notation: 1 < x < 11 Interval Notaion: ( 1,11) x 5 <6 means go a distance of less than 6 units from 5 CASE 2: a is negative Example Algebraic Solution Distance Idea x Always isolate the absolute value term first before solving for x. In this case, first subtract 3 from each side of the equation. We get: x We can stop solving at this point because there is no solution to the inequality. x +4 1 is the same as x ( 4) 1. That is, go a distance of -1 units from 4, but it is not possible to go a negative distance. Exercise 1 Solving Absolute Value Inequality Solve the following inequalities. Write your answer in inequality and interval notation then graph the solution on a number line. a) x 5 12 b) 2x c) x < 2 d) 5 x 1 <8

188 188 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES When solving x a (or x >a), there are 2 possible cases. CASE 1: a is positive x a Inequality Notation: x a or x a Interval Notation: (, a] U [a, ) Number Line: CASE 2: a is negative x a Inequality has every real number as solution because the absolute value graph is always greater than y=a. Inequality Notation: < x < Interval Notation: (, ) The solution to x a (or x < a): CASE 1: When a is positive, the solution is as follows: Inequality Notation: x 1 or x a (or x < a or x >a). Interval Notaion: (, a] U [a, ) (or (,a) U (a, )) CASE 2: When a is negative, the solution is all real numbers. Inequality Notation: < x < Interval Notation: (, )

189 5.2 ABSOLUTE VALUE INEQUALITIES 189 Another way to solve absolute value inequality is to visualize it using the idea of distance. CASE 1: a is positive Example Algebraic Solution Distance Idea x 7 Inequality Notation: x 7 or x 7 Interval Notation: (, 7] U [7, ) x 7 is the same as x 0 7. That is, go a distance of at least 7 units from 0 x +5 >2 Solution: x +5< 2 or x +5>2 x < 7 or x > 3 Inequality Notation: x < 7 or x > 3 Interval Notation: (, 7) U ( 3, ) x + 5 >2 is the same as x ( 5) >2. That is, go a distance that is greater than 2 units from 5 CASE 2: a is negative Example Algebraic Solution Distance Idea Always isolate the absolute value first before solving for x. In this case, first subtract 7 from each side of the equation. x We get: x 9 3. This inequality is always true. Inequality Notation: < x < Interval Notation: (, ) x 9 3 means go a distance that is greater than 3 units from 9. Since distance is always positive, it will always be greater than any negative number.

190 190 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES Problem 1 : Solving Absolute Value Inequality Solve the following inequalitites. Write your answer in inequality and interval notation then graph the solution on a number line. a) x 3 7 b) 2 x +4 >6 c) 1 2x 5 3 d) 1 2x >3 Exercise 2 Solving Absolute Value Inequality Solve the following inequalities. Write your answer in inequality and interval notation then graph the solution on a number line. a) x 5 2 b) 3 2x 4 c) x > 5 d) 2 x 1 < 6

191 5.2 ABSOLUTE VALUE INEQUALITIES 191 Exercise 3 Extension of Concept Below is the graph of the function f (x)= x a) Solve f (x) = 0 for x algebraically and graphically. b) Solve f (x) > 2 algebraically and graphically. Write your answer in both inequality and interval notation. c) Solve f (x) 0 algebraically and graphically. Write your answer in both interval and inequality notaion.

192 192 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES Exercise 1 Application Problem 5.3. ABSOLUTE VALUE APPLICATIONS A meteor will be travelling through the earth's atmosphere. Along its journey, it will be passing by a communicatitons satellite. The meteor's distance, in miles, from the satellite is given by the function D(t) = 10t where t is time in hours. a) Fill the table below. t (time in hours) D(t) (Distance in miles) b) When will the meteor be closer than 25 miles to the satellite? Write an absolute value inequality to model the situation then solve algebraically. c) When will the meteor be at least 30 miles from the satellite? Write an absolute value inequality to model the situation then solve algebraically. d) What is the closest the meteor gets to the satellite? When will that occur?

193 5.3 ABSOLUTE VALUE APPLICATIONS 193 Example 1 Application Problem A woman is 5' 6 tall and weighs approximately 133 lbs. Her weight is desirable and she wants to maintain a healthy weight. Her doctor says that she can deviate from her present weight by at most 15 lbs. and still be considered healthy. Write and solve an absolute value inequality to determine the range of acceptable healthy weight for the woman. Solution. From 133 lbs, the woman can deviate at most 15 lbs. Let w = healthy weight for the woman. To figure out the range of weight that is considered healthy, we solve the inequality w for w. w w w 148 If the woman maintains a weight between 118 and 148 lbs, including 118 and 148 lbs, she will be considered healthy.

194 194 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES Section 5.1: Absolute Value Equations PRACTICE PROBLEMS 1. Solve the following equations. a) x + 2 =6 b) 5 x 6 =10 c) x 2 +1 =2 d) 2x 1 =0 e) 2+ 3x +9= 6 f) 6x =10 g) 2 x 7= 4 h) x =2

195 PRACTICE PROBLEMS 195 Section 5.2: Absolute Value Inequality 2. Solve the following inequalities. Write your solution in inequality and interval notation. Then graph the solution on a number line. a) x 2 b) x 10 c) 2 4+ x > 20 d) 8 + x <1 e) 3 5x 6 0 f) 1 x 4 >2 g) 6x 13 7 h) 2x 3 > 7 i) 2x 7 <11 j) 2x

196 196 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES 3. The graph below shows 2 functions: f (x)= x and g(x)=3. a) Find the x-value(s) for which f (x)= g(x), algebraically and graphically. b) Find the x-values for which f (x) > g(x), algebraically and graphically. Write your answer in both inequality and interval notation. c) Find the x-values for which f (x) g(x), algebraically and graphically. Write your answer in both inequality and interval notation.

197 PRACTICE PROBLEMS 197 Section 5.3: Absolute Value Applications 4. A company produces oatmeal in 18 oz. containers. If the container is underfilled or overfilled by more tha 2% or 0.36 oz, it is sent to the reject pile. Write and solve an absolute value inequality to determine the weights that will go into the reject pile. 5. The ideal body mass to be considered healthy weight is approximately Any deviation from that by at most 3.2 is still considered healthy weight. Write and solve an absolute value inequality to determine the range of body mass for weight to be considered healthy. 6. You are standing at the corner when you hear the siren of an ambulance. Its distance from you, in feet, after you hear the siren is given by the function S(t) = 40t , where t is time in seconds. a) When will the ambulance be closer than 30 feet from you? b) When will the ambulance be at least 55 feet away from you? c) When is the ambulance closest to you?

198 198 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES Chapter 5: Extension 7. Give an example of an absolute value inequality that has a solution 3 x 9. Solve the inequality to verify you have the right solution. 8. Solve the following for x. a) 2x + 1 = x 5 b) x + 3 = 2 x

199 CHAPTER 5 ASSESSMENT 199 CHAPTER 5 ASSESSMENT 1. Solve the following equations. a) 3n+ 2 =14 b) 5x +6 3 =7 c) 2 7y =9 d) 3h = 2 2. Solve the following inequalities. Write your answer in inequality and interval notation. Then graph the solution on a number line. 4 a) 5 4x 9 b) 5 w+6 < 11 c) 2 3 a d) 5 2 y >12

200 200 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES 3. Blood pressure is reported as a ratio of systolic, in mm Hg (millimeters of mercury), over diastolic, in mm Hg. An ideal systolic reading is 120 mm Hg and a healthy blood pressure varies by 20 mm Hg. Write an absolute value inequality and use it to find the range of healthy systolic blood pressure, to ensure neither low blood pressure nor pre-high blood pressure. 4. The graph of f (x)= x 3 4 is shown. a) Find x when f (x) = 0. Solve algebraically and graphically. Write your solution as an ordered pair. b) Find x when f (x) > 3. Solve algebraically and graphically. Write your answer in both inequality and interval notation. c) Find x when f (x) < 1. Solve algebraically and graphically. Write your answer in both inequality and interval notation.

201 CHAPTER 6 POLYNOMIAL EQUATIONS In this chapter, you will learn several methods for solving polynomial equations. Factoring is the first method you will work with to solve polynomial equations. Then our focus shifts to solving quadratic equations when they are not factorable. Two methods will be introduced: completing the square and using the quadratic formula. You will get a taste of something called complex numbers and then we will finish up by putting all the methods together. Pay special attention to the problems you are working on and note details such as signs and coefficients of variable terms. Extra attention to detail will pay off in this chapater. Chapter Topics: Section Factoring Expressions Factoring using the Method of Greatest Common Factor (GCF) Factoring by Trial and Error Section Solving Equations by Factoring Section Completing the Square Section The Quadratic Formula Section Complex Numbers Section Complex Solutions to Quadratic Equations Section Applications of Quadratic Equations 201

202 202 POLYNOMIAL EQUATIONS Component Required? Yes or No Comments Due Score Mini-Lesson Homework Quiz Test Practice Problems Chapter Assessment

203 6.1 FACTORING EXPRESSIONS FACTORING EXPRESSIONS The first method we will discuss in solving some polynomial equations is factoring. Before we jump into this process, we will review the concept of factoring whole numbers. Factoring Whole Numbers To factor the number 60, let us write down a variety of responses, some of which are below: 60 =1 60 (not very interesting but true) 60 = = =3 4 5 All of these are called factorizations of 60, meaning to write 60 as a product of some of the numbers that divide it evenly. The most basic factorization of 60 is as a product of its prime factors (remember that prime numbers are only divisible by themselves and 1). The prime factorization of 60 is: 60= There is only one prime factorization of 60, so we can now say that 60 is completely factored when we write it as 60 = When we factor polynomial expressions, we use a similar process. For example to factor the expression 24x 2, we would first find the prime factorization of 24 and the factors of x = and x 2 =x x Putting these factorizations together, we obtain the following: 24x 2 = x x

204 204 POLYNOMIAL EQUATIONS Let us see how the above information helps us factor more complicated polynomial expressions. Example 1 Factoring Using GCF Method Factor 3x 2 + 6x. Write your answer in completely factored form. Solution. The building blocks of 3x 2 + 6x are the terms 3x 2 and 6x. The factorization of each term is: 3x 2 = 3 x x and 6x = 3 2 x Let us rearrange these factorizations just slightly as follows: 3x 2 = (3 x) x and 6x = (3 x) 2 We can see that (3 x) = 3x is a common factor to both terms. In fact, 3x is the greatest common factor (GCF) of both terms. Let us rewrite the full expression with the terms in factored form and see how that helps us factor the expression. 3x 2 +6x = (3 x) x +(3 x) 2 = (3x) x +(3x) 2 = (3x)(x +2) = 3x(x +2) Always check the factorization by multiplying the final result. 3x(x + 2) = 3x 2 + 6x checks! Problem 1 : Class/Media - Factoring Using GCF Method Factor the following quadratic expressions. Write your answers in completely factored form. a) 11a 2 4a b) 55w 3 25w 2 + 5w

205 6.1 FACTORING EXPRESSIONS 205 Exercise 1 Factoring Using GCF Method Factor the following quadratic expression. Write your answers in completely factored form. a) 64b 2 16b b) 8c 3 20c 2 + 4c Always check to see if there is a greatest common factor before proceeding to use other factoring methods. A list of other possible ways to factor polynomial expressions are shown below. These should be familiar from your previous algebra classes. The following examples and problems will help you review these techniques. Factoring Trinomial of the Form x 2 +bx +c x 2 + bx + c = (x + p)(x +q), where b= p + q and c= p q Factoring Trinomial of the Form ax 2 +bx +c ax 2 +bx + c = (mx + p)(nx + q), where a =m n, c =p q and b =p n+q m Factoring Differences of Squares a 2 b 2 = (a +b) (a b) Note. The sum of squares, a 2 + b 2 is not factorable. It is prime.

206 206 POLYNOMIAL EQUATIONS Example 2 Factoring Using Trial and Error Factor the quadratic expression x 2 + 5x 6. Write your answer in completely factored form. Solution. STEP 1: Look to see if there is a greatest common factor in the expression. If there is, use the GCF method to factor out the greatest common factor. The expression x 2 + 5x 6 has no common factors. STEP 2: For this problem, the middle term is b=5 and the constant term is c= 6. We need to identify p and q. These will be two numbers whose product is c = 6 and sum is b = 5. One way to do this is to list different numbers whose product is 6, then see which pair has a sum of 5. Product = 6 Sum = No 3 2 No 1 6 YES 1 6 No STEP 3: Write in factored form. x 2 +5x 6 = (x 1) (x + 6) STEP 4: Check by multiplying the factors. (x 1)(x +6) = x 2 +6x x 6 = x 2 +5x 6 Check! Problem 2 : Class/Media - Factor the Expression Factor each of the following expressions completely. Check your answers. a) a 2 + 7a+ 12 b) 5w 3 +5w c) x d) 6x 2 x 15

207 6.1 FACTORING EXPRESSIONS 207 Exercise 2 Factor the expression Factor each of the following expressions completely. Check your answers. a) 4x 2 + 2x 6 b) m 3 m 2 30 c) 6r r 5 d) 81 y 4 e) n 4 + 8n 2 9 f) m 3 + 3m 2 4m 12 Factoring Sum and Differences of Cubes a 3 +b 3 =(a +b) (a 2 ab+ b 2 ) - Sum of Cubes a 3 b 3 =(a b) (a 2 + ab+ b 2 ) - Difference of Cubes Note. The trinomial (a 2 ab +b 2 ) and (a 2 +ab+ b 2 ) cannot be factored further. Note. For cubes, their sum and differences are factorable. However, for squares, the difference is factorable but not the sum. That is, a 2 b 2 =(a +b) (a b) but a 2 + b 2 is prime.

208 208 POLYNOMIAL EQUATIONS Example 3 Factoring Cubes Factor the following expressions completely. a) x b) 27x 3 1 Solution. a) Rewrite x 3 +8 as the sum of cubes: x Let a=x and b=2. Then use the sum of cubes formula to factor. x 3 +8 = x = (x +2) (x 2 2x ) = (x +2) (x 2 2x + 4) b) Rewrite 27x 3 1 as the difference of cubes: (3x) Let a = 3x and b = 1. Then use the difference of cubes formula to factor. 27x 3 1 = (3x) = (3x 1) [(3x) 2 + (3x) (1)+ 1 2 ] = (3x 1) (9x 2 + 3x + 1) Exercise 3 Factoring Cubes Factor the following expressions completely. a) x 3 64 b) m c) 1+ 8n 3 d) 27 64a 3

209 6.2 SOLVING EQUATIONS BY FACTORING SOLVING EQUATIONS BY FACTORING In this section, we will see how a polynomial equation can be solved algebraically using factoring methods. In order to do this we must recall a very simple, but useful, fact about the product of numbers: the zero product property. The Zero Product Property: If a b = 0, then a=0 or b = 0 To solve a polynomial equation by factoring: Step 1: Make sure the equation is set to 0. Step 2: Completely factor the polynomial expression. Step 3: Apply the Zero Product Property. Set each linear factor equal to 0 and solve for x. Step 4: Verify that the solution solves the equation. Example 1 Solving Equations by Factoring Solve the following equations by factoring. Be sure to check your answers. a) 5x 2 = 10x b) x 2 7x + 12= 2 Solution. a) Solve 5x 2 = 10x for x. Step 1: Make sure equation is set to 0. Subtract 10x from each side of the equation to get: 5x 2 10x = 0. Note. You cannot divide each side by 5x unless x 0. If you divide by 5x, you may be dividing by zero. This can lead to undefined terms. Step 2: Completely factor the polynomial expression. First, check if there is a greatest common factor. Yes there is! 5x is common to both terms. Write the polynomial expression in a completely factored form. 5x 2 10x = 0 5x(x 2) = 0

210 210 POLYNOMIAL EQUATIONS Step 3: Set each linear factor equal to 0 and solve for x. 5x = 0 or x 2= 0 x = 0 or x = 2 Step 4: Verify that the each solution solves the equation. Replace one solution at a time. When x = 0: When x =2: 5(0) 2 =? 10(0) 5(2) 2 =? 10(2) 5(0) = 0 5(4) = 20 b) Solve x 2 7x + 12= 2 for x. Step 1: Make sure equation is set to 0. Subtract 2 from each side of the equation to get: x 2 7x + 10= 0. Step 2: Completely factor the polynomial expression. First, check if there is a greatest common factor. There is no greatest common factor. Next, factor the trinomial. x 2 7x +10 = 0 (x 5)(x 2) = 0 Step 3: Set each linear factor equal to 0 and solve for x. x 5= 0 or x 2= 0 x = 5 or x = 2 Step 4: Verify that the each solution solves the equation. Replace one solution at a time. When x = 5: When x =2: (5) 2 7(5)+ 12 =? 2 (2) 2 7(2) +12 =? = = 2

211 6.2 SOLVING EQUATIONS BY FACTORING 211 Problem 1 : Class/Media - Solving Equations by Factoring Solve each equation by factoring. Show all of your work and be sure to check your answers. a) (4x 5)(x +2) =0 b) 6w 2 = 8w c) y 2 + 5y = y 3 d) p(3p + 2)= 5 Exercise 1 Solving Equations by Factoring Solve each equation by factoring. Show all your work and be sure to check your answers. a) 4w 2 = 100 b) p(p 2 +4p) =5p c) x 4 13x = 0 d) (2y 1) (y 3)= 7

212 212 POLYNOMIAL EQUATIONS Example 2 Solving for a Variable in a Formula Solve for the indicated variable. Simplify your answer. a) A= P + Prt for t b) A =P +Prt for P Solution. a) A= P + Prt for t Subtract P from each side Divide each side by Pr Therefore, t = A P Pr A P = Prt A P = t Pr b) A= P + Prt for P Factor P first Divide each side by (1+ rt) Therefore, P = A 1 +rt A = P(1+ rt) A = P 1+ rt Exercise 2 Solving for a Variable in a Formula Solve for the indicated variable. Simplify your answer. a) A= 1 Bh for B b) C =GmT + T for T 2 c) ab+ ac =bc for c d) tv + 2rt =2V for V

213 6.3 COMPLETING THE SQUARE COMPLETING THE SQUARE We will now focus on a specific polynomial equation called the Quadratic Equation. A quadratic equation is in standard form if the equation is of the form ax 2 + bx + c = 0, where a 0. Sometimes, quadratic equations are factorable. Often, they are not. Two other techniques for solving quadratic equations will be discussed. They are: Completing the Square and the Quadratic Formula, which is also based on the idea of completing the square. Let's recall first how to deal with simple quadratic equations. Suppose we have a quadratic equation of the form x 2 =k, where k 0. To solve for x, we take the square root of each side of the equation: x 2 = k p p x 2 = k p x = k p p x = k or x = k p In short, we can write the solution as: x = ± k. p Square Root Property states that if x 2 =k, where k 0, then x =± k. p p Note. Remember that x 2 = x. Writing x 2 =x is incorrect. p p Also take note that k ± k. p p p For example, 4 ±2. However, 4 = 2 and 4 = 2. Example 1 Square Root Property Solve x 2 = 4 for x. Solution. Given Problem x 2 = 4 p p Take the square root of each side x 2 = 4 p p Remember x 2 = x and 4 =2 x = 2 Recall how to solve absolute value equations x = 2 or x =2 Check: when x = 2 Check: when x = 2 ( 2) 2 = 4 (2) 2 = 4

214 214 POLYNOMIAL EQUATIONS Example 2 Square Root Property Solve the following quadratic equations using the square root property. Write your answer in exact form. Be sure to simplify all answers. If the answer is irrational, approximate to 1 decimal place. a) (2x 1) 2 = 3 b) 2(x 5) 2 = 18 Solution. a) Since the square term is already isolated, we can apply the square root property right away. (2x 1) 2 = 3 p (2x 1) 2 = 3 p 2x 1 = 3 p 2x 1 = ± 3 p 2x = 1± 3 x = 1± p 3 (Irrational Answers) 2 x 1.4 or 0.4 (Approximated to 1 decimal place) b) The square term is not isolated. Divide each side by 2 before using the square root property. 2(x 5) 2 = 18 (x 5) 2 = 9 p (x 5) 2 = 9 x 5 = 3 x 5 = ±3 x = 5± 3 x = 5+ 3 or x = 5 3 x = 8 or x = 2

215 6.3 COMPLETING THE SQUARE 215 Exercise 1 Square Root Property Solve the following quadratic equations by the square root property. Write your answer in exact form. Be sure to simplify all answers. If answers are irrational, approximate to 3 decimal places. a) x 2 = 25 b) 3x 2 = 6 c) (x 1) 2 = 7 d) (2x + 3) 2 9= 7 Completing the Square can be used to solve quadratic equations written in standard form, ax 2 +bx +c=0, where a 0. To solve a Quadratic Equation by completing the square: Step 1: Move the constant to one side of the equation. Step 2: Make sure a which is the coefficient of x 2 is 1. If not, divide the entire equation by a. Step 3: Take b which is the coefficient of x and divide it by 2. Square the answer and add it to each side of the equation. Step 4: Factor perfect square trinomial expression and combine terms on the other side of the equation. Step 5: Apply the Square Root Property. Step 6: Solve for x.

216 216 POLYNOMIAL EQUATIONS Example 3 Solve Quadratic Equation by Completing the Square Solve the following quadratic equations by completing the square. Give answers in exact form. Be sure to simplify all answers. If the answers are irrational, approximate to 2 decimal places. a) x 2 8x + 3= 0 b) 2x 2 + 6x 1= 0 Solution. a) Given problem x 2 8x +3 = 0 Step 1: Move constant to right-hand side of equation x 2 8x = 3 Step 2: Coefficient of x 2 is already 1 Step 3: Take coefficient of x, which is 8 Divide 8 by 2, to get 4 Square 4: ( 4) 2 =16 Add 16 to each side of equation x 2 8x +16 = Step 4: Factor left-side; combine terms on right-side (x 4) 2 = 13 Step 5: Apply Square Root Property (x 4) 2 = p 13 p Step 6: Solve for x x 4 = 13 p x 4 = ± 13 p Exact Answers (Irrational) x = 4± 13 Approximated Answers x 7.61 or 0.39

217 6.3 COMPLETING THE SQUARE 217 b) Given problem 2x 2 + 6x 1 = 0 Step 1: Move constant to right-hand side of equation 2x 2 + 6x = 1 Step 2: Coefficient of x 2 is not 1 Step 3: Take coefficient of x, which is 3 Divide each side by 2 x 2 + 3x = 1 2 Divide 3 by 2, to get 3 2 Square : 3 = 9 ( 2) 4 Add 9 4 to each side of equation x2 + 3x = Step 4: Factor left-side; combine terms on right-side ( x + 3 2) ( x + 3 2) 2 2 = = 11 4 Step 5: Apply Square Root Property 2 ( x + 3 2) = 11 4 Step 6: 3 Solve for x x + = x + 3 p 2 = ± 11 2 p Exact Answers (Irrational) x = 3 2 ± 11 2 or p 3± 11 x = 2 Approximated Answers x 0.15 or 3.15

218 218 POLYNOMIAL EQUATIONS Exercise 2 Solve Quadratic Equaations by Completing the Square Solve the following quadratic equations by completing the square. Give answers in exact form. Be sure to simplify all answers. If the answers are irrational, approximate to 2 decimal places. a) x 2 2x 5= 0 b) x 2 + 8x = 9 c) 3x 2 + 9x = 6 d) 2x 2 5x 3= 0

219 6.4 THE QUADRATIC FORMULA THE QUADRATIC FORMULA The Quadratic Formula can be used to solve quadratic equations written in standard form, ax 2 +bx + c= 0, where a 0. The Quadratic Formula: x = b± p b2 4ac 2a To solve a Quadratic Equation using the Quadratic Formula: Step 1: Make sure the quadratic equation is in standard form, ax 2 +bx + c= 0. Step 2: Identify the coefficients a, b, and c. Step 3: Substitute these values into the Quadratic Formula. Step 4: Simplify your result completely. Step 5: Verify that the solution solves the equation. Do you ever wonder where this formula came from? Well, you can actually derive this formula directly from the quadratic equation in standard form, ax 2 + bx + c = 0 by Completing the Square. We will go through the derivation by going through the steps in completing the square. Deriving the Quadratic Formula from ax 2 + bx +c=0: Quadratic Equation in Standard Form ax 2 +bx +c = 0 Step 1: Subtract c from each side of equation ax 2 +bx = c Step 2: Coefficient of x 2 is not 1 Divide each side by a x 2 + b a x = c a

220 220 POLYNOMIAL EQUATIONS Step 3: Take coefficient of x, which is b a Divide b a by 2 (or multiply b a by 1 2 ), to get b 2a Square b 2a : b 2= b2 ( 2a) 4a 2 Add b2 4a2 to each side of equation x2 + b a x + b2 4a 2 = c a + b2 4a 2 Step 4: Factor left-side; combine terms on right-side ( x + b 2a) 2 = 4ac 4a 2 + b2 4a 2 Step 5: Apply Square Root Property ( x + b 2a) 2 = 4ac+ b 2 4a 2 Step 6: ( x + b 2a) ( x + b 2a) 2 2 = b 2 4ac 4a 2 p = b 2 4ac 2a b p Solve for x x + = b 2 4ac 2a 2a x + b p 2a = ± b2 4ac 2a p x = b 2a ± b2 4ac 2a The Quadratic Formula x = b ± p b2 4ac 2a

221 6.4 THE QUADRATIC FORMULA 221 Example 1 Solve Quadratic Equations Using the Quadratic Formula Solve 3x 2 2 = x for x using the quadratic formula. Give answer in exact form. If answer is irrational, approximate to 3 decimal places. Verify your result. Solution. Given problem 3x 2 2 = x Step 1: Get equation in standard form by subtracting x from each side 3x 2 2 x = 0 Rewrite polynomial expression in descending order 3x 2 x 2 = 0 Step 2: Identify a= 3, b = 1, c = 2 Step 3: Substitute the values of a, b and c into quadratic formula x = ( 1) ± ( 1)2 4(3) ( 2) 2(3) = 1 ± = 1 ± 25 6 = 1 ±5 6 Step 4: Simplify result completely x = x = 6 6 or x = or x = 6 x =1 or x = 2 3 Step 5: Verify solution solves equation When x = 2 3 : 3 ( 2 3) 2 2 =? 2 3 3( 4 2 = 9)? = 2 3 When x =1: 3(1) 2 2=? 1 3(1) 2=? 1 3(1) 2 2= 1 Final solution: x =1, x = 2 3

222 222 POLYNOMIAL EQUATIONS Problem 1 : Class/Media - Solve Quadratic Equations Using the Quadratic Formula Solve the following using the quadratic formula. Give answers in exact form. If answer is irrational, approximate to 3 decimal places. a) x 2 + 3x + 10= 0 b) 2x 2 4x =3 Exercise 1 Solve Quadratic Equations Using the Quadratic Formula Solve 3x 2 = 7x + 2 for x using the quadratic formula. Give answers in exact form. If answer is irrational, approximate to 2 decimal places. Verify your result.

223 6.5 COMPLEX NUMBERS COMPLEX NUMBERS Suppose we are asked to solve the quadratic equation x 2 = 1. Well, right away you should think that this looks a little weird. If I take any real number times itself, the result is always positive. Therefore, there is no real number x such that x 2 = 1. Hmmm.... well, let us apporach this using the Quadratic Formula and see what happens. To solve x 2 = 1, we write it in standard form as x 2 +1=0. Thus, a= 1, b= 0, and c= 1. Entering these into the quadratic formula, we get the following: 0± (0) 2 4(1)(1) x = 2(1) = ± p 4 2 = ± p 4( 1) 2 = ± p 4 p 1 p2 ±2 1 = p 2 = ± 1 p p Well, again, the number 1 does not live in the real number system nor does the number 1. Yet these are the two solutions to our equation x 2 + 1= 0. The way mathematicians have handled this problem is to define a number system that is an extension of the real number system. This system is called the Complex Number System and it has, as its base defining characteristic, that equations such as x = 0 can be solved in this system. To do so, a special definition is used and that is the definition that: p i = 1 With this definition, the solutions to x 2 + 1=0 are x = i and x = i. When will we see these kinds of solutions? We will see solutions that involve the complex number i when we solve quadratic equations whose solutions involve negative radicands. The following examples will help you understand how to work with these new concepts. Complex Numbers a + bi Complex numbers are an extension of the real number system. The Standard form for a complex number is: p a +bi, where a and b are real numbers and i = 1. We call a, the real part and b, the imaginary part.

224 224 POLYNOMIAL EQUATIONS Example 1 Complex Numbers Simplify each of the following and write your answer in the form a+bi. p p p a) 9 b) 7 c) 2 Solution. a) b) c) p 9 p 7 p p p = 9 1 = 3i p p = 7 1 p p = 7 i or i 7 = 3 + p 49 p 1 2 = 3 +7i 2 = i NOTATION. The imaginary number, i, is often written in front of the radical. That is, 3i is often written as i 3 so it is clear that i is not inside the radical sign. Exercise 1 Complex Numbers Simplify each of the following and write your answer in the form a+bi. p p a) 5 b) p c) 4 12 d) p

225 6.5 COMPLEX NUMBERS 225 Complex Numbers p All numbers of the form a+bi, where a and b are real numbers and i = 1 Examples: 3 +4i, i, 7, 8i 5 Real Numbers Include all RATIONAL and IRRATIONAL NUMBERS Rational Numbers Ratios of Integers Decimals that terminate or repeat Examples: 0.50 = 1 2, 0.75= 3 4, 1 3 = Irrational Numbers Decimal representations for these numbers never terminate and never repeat Examples: p p p 2 π, e, 7, 3 11, 5 Integers Include Zero, Counting Numbers and their negatives: {, 5, 4, 3, 2, 1, 0, 1, 2, 3,4, 5, } Whole Numbers Include Counting Numbers and Zero {0, 1,2, 3,4, 5, } Counting Numbers {1,2, 3,4, 5,6, 7, }

226 226 POLYNOMIAL EQUATIONS THE COMPLEX NUMBER SYSTEM Complex numbers are an extension of the real number system. As such, we can perform operations on complex numbers. This includes addition, subtraction, multiplication and powers. p A complex number is written in the form a+bi, p where a and b are real numbers and i = 1. Extending this definition a bit, we define i 2 =( 1) 2 = 1. p p p Note. The multiplication property, a b = a b does NOT apply if both both a <0 and b< 0. p p p p That is, i 2 =( 1) 2 = =1. It is a good idea to write p the term with a negative radicand in complex form immediately. For example, if you have 9, rewrite it as 3i. Example 2 Operations with Complex Numbers Perform the indicated operations. Simplify and write answers in the form a+bi. p p a) 9 4 b) (8 5i)+ (1+ i) c) (3 2i) (4+ i) e) (2+ i)(4 2i) d) 5i(8 3i) f) (3 5i) 2 Solution. a) Since both radicands are negative, we cannot use the multiplication property for radicals. p p p p p p That is, 9 4 ( 9) ( 4) = 36 = 6. Instead, rewrite 9 = 3i and 4 = 2i. p p 9 4 = (3i) (2i) b) = 6i 2 (Recall i 2 = 1) = 6( 1) = 6 (8 5i) +(1 +i) = 8 5i +1+i = 9 4i

227 6.5 COMPLEX NUMBERS 227 c) (3 2i) (4 +i) = 3 2i 4 i = 1 3i d) 5i(8 3i) = 40i 15i 2 (Recall i 2 = 1) = 40i 15( 1) = 40i+ 15 = i e) (2 +i)(4 2i) = 8 4i +4i 2i 2 = 8 2i 2 = 8 2( 1) = 8 +2 =10 f) (3 5i) 2 = (3 5i)(3 5i) = 9 15i 15i+ 25i 2 = 9 30i+ 25( 1) = 9 30i 25 = 16 30i Exercise 2 Working with Complex Numbers Simplify each of the following and write your answer in the form a+bi. p 15 9 a) b) (10+ 4i)(8 5i) 3

228 228 POLYNOMIAL EQUATIONS Example 3 Verifying Complex Solutions Verify that x = 4+ 3i is a solution to the quadratic equation x 2 8x + 25= 0. Solution. Substitute x = 4+ 3i into the given quadratic equation. Remember i 2 = 1. (4+ 3i) 2 8(4+ 3i)+ 25 =? 0 (4+ 3i)(4+ 3i) 8(4+ 3i)+ 25 =? i +12i +9i i+ 25 =? i 2 =? ( 1) =? = 0 Therefore, x = 4+ 3i is a solution to the quadratic equation x 2 8x + 25= 0. Exercise 3 Verifying Complex Solutions Verify that the given value of x is a solution to the given quadratic equation. a) 5x 2 = 20 where x = 2i b) x 2 2x + 2= 0 where x = 1+ i

229 6.6 COMPLEX SOLUTIONS TO QUADRATIC EQUATIONS COMPLEX SOLUTIONS TO QUADRATIC EQUATIONS Example 1 Solving Quadratic Equations with Complex Solutions Solve 2x 2 + x + 1= 0 for x. Leave results in the form a+bi. Solution. The graph to the right shows the function f (x)= 2x 2 + x + 1. Notice that that the graph does not cross the x-axis (or y = 0) at all. This is an example where the result has no Real Number Solutions but two unique Complex Number Solutions. We will use the quadratic formula to solve the equation. First, make sure the equation is in standard form, 2x 2 + x + 1 = 0. Then, identify the coefficients a = 2, b = 1, c = 1. Substitute the values into the quadratic formula and simplify as follows: x = 1± (1)2 4(2)(1) 2(2) 1± 1 8 = 4 1± 7 = 4 = 1± i 7 4 = 1 4 ± 7 4 i The final solutions are: x 1 = i and x 2 = i

230 230 POLYNOMIAL EQUATIONS Problem 1 : Class/Media - Solving Quadratic Equations with Complex Solutions Solve x 2 + 4x + 8= 1 for x. Leave results in the form a+bi. Exercise 1 Solving Quadratic Equations with Complex Solutions Solve 2x 2 3x = 5 for x. Leave results in the form a+bi.

231 6.7 APPLICATIONS OF QUADRATIC EQUATIONS APPLICATIONS OF QUADRATIC EQUATIONS Work through the following problems to put the solution methods of factoring, completing the square and using the quadratic formula together while working with the same equation. Exercise 1 Solving Quadratic Equations Given the quadratic equation x 2 + 3x 7= 3, solve using the process indicated below. a) Solve by factoring. Show all steps and clearly identify your final solutions. b) Solve by completing the square. Show all steps and clearly identify your final solutions. c) Solve using the quadratic formula. Show all steps and clearly identify your final solutions.

232 232 POLYNOMIAL EQUATIONS As seen from the above example, there are different ways to solve a quadratic equation. There is no one right method, unless the equation is not factorable. However, some methods may be easier than others. Use your best judgment. We will now apply solving quadratic equations to application problems. Example 1 Application Problem Bacillus thuringiensis (Bt) is a naturally occurring soil bacterium that produces a toxin that is deadly to some insects, including moths and butterflies. Scientists have more recently modified some crops, such as corn, to allow the modified Bt crops to produce their own Bt toxin. However, studies have documented negative effects of Bt crops on other non-target species, particularly the Monarch Butterfly. A possible solution to reduce the risks of Bt crops to non-target species is the creation of buffer zones around crop fields. It is suggested that between 20% to 30% of the crop planted be non-bt varieties to serve as refuge. (Source: John Obrycki and Laura Jesse from the Leopold Center for Sustainable Agriculture at Iowa State University) Suppose a farmer has an 80-meter by 100-meter rectangular plot of land. He wants to use part of the land to plant Bt corn with a 20% buffer zone. The buffer zone is to be of identical width around the entire field. Find the width of the buffer zone and the dimension of the Bt cornfield. Be sure to include units in your answers. (Round final answers to 2 decimal places) Solution. Area of land the farmer has: (80 m) (100 m) = 8000 sq. meters Area ofbuffer zone: 20% of 8000 sq. meters = (0.20) (8000) = 1600 sq. meters Let x = width of the buffer zone

233 6.7 APPLICATIONS OF QUADRATIC EQUATIONS 233 Area of Bt Crop Field + Area of Buffer Zone = Total Land Area (80 2x) (100 2x) = x 200x + 4x = x 2 360x = x 2 360x = 0 x 2 90x = 0 Solve for x. Note equation is not factorable. Therefore, solve by either completing the square or using the quadratic formula. We will use the quadratic formula for this problem. Identify a = 1, b = 90 and c = 400. x = ( 90)± ( 90)2 4(1)(400) 2(1) 90± = 2 = 90± ± = 2 = 45± 5 65 We have 2 solutions: x = meters and x = meters Check to see if our solutions make sense. Having a width of meters for the buffer zone is impossible because one side of the plot is only 80 meters wide. Therefore, the width of the buffer zone must be 4.69 meters. The area of the Bt cornfield: (80 2(4.69))(100 2(4.69))=(7.06)(9.06) 6400 square meters

234 234 POLYNOMIAL EQUATIONS Exercise 2 Application Problem A gardener has a 4-meter by 5-meter rectangular plot of land. She wants to convert 30% of the land into a decorative walkway and the rest to plant flowers. The decorative walkway is to be of identical width around the plot of land. Find the width of the decorative walkway and the dimension of the plant area. (Round answers to 2 decimal places)

235 6.7 APPLICATIONS OF QUADRATIC EQUATIONS 235 Exercise 3 Application Problem You are helping your neighbor grow lettuce. She wants it to be colorful and easy to grow, so you choose the following: Red Fire Leaf, Tom Thumb, Speckles and Mottistone. Her plot is 255 ft 2. a) Determine the equation for the area of your neighbor's property. b) What length and width will the property need to be? Show all your work. c) Determine the area of each section of land. Include units in your answer. Red Fire Leaf: Tom Thumb: Mottistone: Speckles:

236 236 POLYNOMIAL EQUATIONS Section 6.1: Factoring Expressions PRACTICE PROBLEMS 1. Factor each of the following expressions completely. a) a 2 + 7a+ 6 b) 3x 3 +12x c) m 4 81 d) 2w 2 13w +11 e) 5y 2 11y + 2 f) 3x 2 +12x +9 g) 36 k 2 h) 6p 2 +p 1 i) n 3 4n 2 n+ 4 j) 3c 2 6c +24

237 PRACTICE PROBLEMS 237 Section 6.2: Solving Equations by Factoring 2. Solve each of the following quadratic equations by factoring. a) 2x 2 = 8x b) 2x 2 =8 c) 3x x + 36= 0 d) x 2 42 =x e) (x + 2)(x 6) =9 f) 49 x 2 =0 g) x 4 5x 2 + 4= 0 h) 3x 3 +5x 2 2x =0 i) 2x 2 + 3x = 20 j) 4x 3 +4x 2 =x +1

238 238 POLYNOMIAL EQUATIONS 3. Solve for the indicated variable. Simplify all answers a) V = 1 BH for B b) Ax +By =C for y 3 c) V = Ir + IR for I d) AB AC =C for C e) RR 1 + RR 2 =R 1 R 2 for R f) S= 2lw +2hw +2lh for h Section 6.3: Completing the Square 4. Solve each quadratic equation using the square root property. Simplify all answers. If solutions are irrational, approximate to 2 decimal places. a) x 2 = 45 b) 2x 2 = 32 c) (2x 3) 2 = 15 d) 6(x 5) 2 3= 21

239 PRACTICE PROBLEMS Solve each quadratic equation by completing the square. Simplify all answers. If solutions are irrational, approximate to 2 decimal places. a) x 2 4x 7= 0 b) 4x 2 8x = 12 c) 2x 2 + 6x 8= 0 d) x 2 5x = 1 Section 6.4: The Quadratic Formula 6. Solve x 2 x 2= 0 using the quadratic formula by filling in the blanks. Be sure to check your answers. ( ) ± ( ) 2 4( ) ( ) x = 2( ) x = ( ) ± p ( ) ( ) ( ) x = ( ) ± p ( ) ( ) x 1 = ( ) + p ( ) ( ) x 1 = ( ) +( ) ( ) x 1 = ( ) ( ) or x 2 = ( )+ p ( ) ( ) or x 2 = ( )+ ( ) ( ) or x 2 = ( ) ( ) x 1 = or x 2 =

240 240 POLYNOMIAL EQUATIONS 7. Solve each quadratic equation by using the quadratic formula. a) 2x 2 5x 4= 0 b) 4x 2 = 2x + 6 c) 4x + 1= 6x 2 d) 12= 3x + 2x 2 Section 6.5: Complex Numbers 8. Verify that the given value of x is a solution to the given equation. a) x 2 + 9= 0 where x = 3i b) x 2 2x = 5 where x = 1+ 2i

241 PRACTICE PROBLEMS Simplify each of the following. Write answers in the form a+bi. p p a) 81 b) 11 c) (4 2i) (6+ 8i) d) 3i(2 4i) e) (3 i)(2 +i) f) (4 8i) 3(4 +4i) g) (2+ i) 2 h) p i) p j) p 2 4 4(2)(5) 4

242 242 POLYNOMIAL EQUATIONS Section 6.6: Complex Solutions to Quadratic Equations 10. Solve the following quadratic equations in the complex number system. Leave your final solution in the complex form, a+bi. 1 a) 2 x2 + 5x + 17 = 0 b) x 2 + 2x = 5 c) 4x 2 + 9= 0 d) 4x = 3x 2 + 7

243 PRACTICE PROBLEMS 243 Section 6.7: Applications of Quadratic Equations 11. Farmer Treeman wants to plant four crops on his land: Corn, Cucumber, Kale and Collard. He has 40,000 square feet for planting. He needs the length and width of the property to be as shown in the picture below (measured in feet). a) Determine the equation for the area of the farmer's property. b) What will the length and width of the property need to be? Show work. c) Determine the area of each section of the land. Include units in your answers. Corn: Kale: Cucumber: Collard:

244 244 POLYNOMIAL EQUATIONS 12. A rectangular plot of land is 15 feet by 20 feet. 3 of the land is going to be torn up to make 5 room for a swimming pool. The rest is for a decorative stone walkway surrounding the pool that is of uniform width. a) What is the area of the pool? b) How wide is the decorative stone walkway? (Round answer to 2 decimal places) Chapter 6: Extension 13. Give a polynomial function with solutions at the following values of x. a) x = 3 and x = 2 b) x = 0 and x = 1 2 c) x = 3 2 and x =4 d) x = 1,x =1 and x =2