Linear Functions, Equations, and Inequalities

Transcription

1 CHAPTER Linear Functions, Equations, and Inequalities Inventory is the list of items that businesses stock in stores and warehouses to supply customers. Businesses in the United States keep about.5 trillion dollars worth of goods in inventory. You will use linear functions to manage the inventory levels of a business.. Tanks a Lot Introduction to Linear Functions p. 3.2 Calculating Answers Solving Linear Equations and Linear Inequalities in One Variable p..3 Running a 0K Slope Intercept Form of Linear Functions p. 2.4 Pump It Up Standard Form of Linear Functions p Shifts and Flips Basic Functions and Linear Transformations p Inventory and Sand Determining the Equations of Linear Functions p Absolutely! Absolute Value Equations and Inequalities p Inverses and Pieces Functional Notation, Inverses, and Piecewise Functions p. 67 Chapter Linear Functions, Equations, and Inequalities

2 Mathematical Representations INTRODUCTION Mathematics is a human invention, developed as people encountered problems that they could not solve. For instance, when people first began to accumulate possessions, they needed to answer questions such as: How many? How many more? How many less? People responded by developing the concepts of numbers and counting. Mathematics made a huge leap when people began using symbols to represent numbers. The first numerals were probably tally marks used to count weapons, livestock, or food. As society grew more complex, people needed to answer questions such as: Who has more? How much does each person get? If there are 5 members in my family, 6 in your family, and 0 in another family, how can each person receive the same amount? During this course, we will solve problems and work with many different representations of mathematical concepts, ideas, and processes to better understand our world. The following processes can help you solve problems. Discuss to Understand Read the problem carefully. What is the context of the problem? Do you understand it? What is the question that you are being asked? Does it make sense? Think for Yourself Do I need any additional information to answer the question? Is this problem similar to some other problem that I know? How can I represent the problem using a picture, a diagram, symbols, or some other representation? Work with Your Partner How did you do the problem? Show me your representation. This is the way I thought about the problem how did you think about it? What else do we need to solve the problem? Does our reasoning and our answer make sense to one another? Work with Your Group Show me your representation. This is the way I thought about the problem how did you think about it? What else do we need to solve the problem? Does our reasoning and our answer make sense to one another? How can we explain our solution to one another? To the class? Share with the Class Here is our solution and how we solved it. We could only get this far with our solution. How can we finish? Could we have used a different strategy to solve the problem? 2 Chapter Linear Functions, Equations, and Inequalities

3 . Tanks a Lot Introduction to Linear Functions Objectives In this lesson, you will: Define dependent and independent variables. Write linear functions. Graph linear functions. Use multiple representations of linear functions to model and solve problems. Key Terms variable independent variable dependent variable function linear function Problem An oil storage tank farm has an empty tank with a capacity of 5000 gallons. It will be filled with motor oil using a pipe that fills the tank at the rate of twelve gallons per minute.. How much motor oil will be in the tank after 20 minutes? One hour? Two hours? 2. After how many minutes will there be exactly 000 gallons of motor oil in the tank? 3. In this problem, there are two quantities that are changing. What are they? One quantity, the dependent quantity, depends on the other, the independent quantity. a. Independent quantity: b. Dependent quantity: Lesson. Introduction to Linear Functions 3

4 4. Assign a variable to each of the quantities and write an equation that shows the relationship between these variables. The variable assigned to the independent quantity is called the independent variable, and the variable assigned to the dependent quantity is called the dependent variable. 5. Use the answers to Questions through 4 to complete the following table to identify the quantities that are changing, the units that are used to measure these quantities, and the expressions representing each of these quantities. Then construct a graph. Quantity Name Unit Expression 4 Chapter Linear Functions, Equations, and Inequalities

5 6. How would you describe the graph of this situation? 7. Use your equation to determine how much oil is in the tank after two thirds of an hour. 8. Use your equation to solve for the number of minutes it would take to have exactly 2234 gallons of oil in the tank. 9. How much motor oil is added to the tank every minute? In 0 minutes? In one hour? 0. For what interval of time will there be less than 000 gallons of oil in the tank?. Calculate the amount of time it takes to fill an empty tank. How did you get your answer? Lesson. Introduction to Linear Functions 5

6 2. Choose different values from two rows in your table in Question 5. Solve for the change in the dependent variable and the change in the independent variable. Calculate the quotient obtained when the change in the dependent variable is divided by the change in the independent variable. 3. Compare your answer with the other students in your group, and then compare your answer with another group. Explain all similarities and differences. Problem 2 A second tank that is currently full contains 20,000 gallons of home heating oil. This tank will be emptied through a drain pipe at the rate of twenty five gallons per minute.. How much heating oil will be in the tank after 0 minutes? One hour? Ten hours? 2. After how many minutes will there be exactly 0,000 gallons of heating oil in the tank? 3. In this problem, there are two quantities that are changing. What are they? a. Independent quantity: b. Dependent quantity: 6 Chapter Linear Functions, Equations, and Inequalities

7 4. Assign a variable to each of the quantities and write an equation that shows the relationship between these variables. a. What does 20,000 represent in this equation with respect to the problem situation? b. What does 25 represent in this equation with respect to the problem situation? Use the answers to Questions through 4 to complete the table. Then construct a graph. Quantity Name Unit Expression Lesson. Introduction to Linear Functions 7

8 5. How would you describe the graph of this situation? 6. Use your equation to calculate the amount of oil remaining at exactly 37 minutes into emptying the tank. 7. Use your equation to calculate the number of minutes it would take for there to be exactly 675 gallons of oil left in the tank. 8. Calculate the total amount of time it takes to empty the tank. Explain how you got your answer. 9. For what interval of time will there be more than 0,000 gallons but less than 5,000 gallons of oil in the tank? 0. For every minute, how much heating oil is drained from the tank? In 0 minutes? An hour? 8 Chapter Linear Functions, Equations, and Inequalities

9 . Choose different values from two rows in your table. Solve for the change in the dependent variable. Solve for the change in the independent variable. Calculate the quotient obtained when the change in the dependent variable is divided by the change in the independent variable. Remember A function is a relation that maps each value of the independent variable to one and only one value of the dependent variable. 2. Compare your answer with the other students in your group, and then compare your answer with another group. Explain all similarities and differences. 3. Problems and 2 are examples of mathematical relations called linear functions. What about their graphs indicates this? 4. List at least three characteristics of linear functions. Be prepared to share your work with another pair, group, or the entire class. Lesson. Introduction to Linear Functions 9

10 0 Chapter Linear Functions, Equations, and Inequalities

11 .2 Calculating Answers Solving Linear Equations and Linear Inequalities in One Variable Objectives In this lesson, you will: Solve one-step and two-step linear equations. Solve and graph linear inequalities. Key Terms transformations simplification inequality number line Problem When you wrote an equation for the situation in Problem in Tanks a Lot, you defined variables to represent the quantities that were changing and used those variables to write an equation that showed the relationship between the quantities. You then used the equation to answer a number of questions about the situation. Use that same equation to find out how long it will take for the tank to contain 200 gallons of motor oil. When you have an equation with two variables, and you substitute a number for one of the variables, you are left with a linear equation to be solved for the remaining variable. To solve an equation of this type, there are four basic transformations that you can apply to both sides of the equation: Addition Subtraction Multiplication Division Lesson.2 Solving Linear Equations and Linear Inequalities in One Variable

12 To solve the equation, you also have to perform a simplification. The simplifications most commonly used are: Add/Subtract terms Perform multiplication Simplify fractions Simplify signs Distribute In each of the following equations, perform the transformation and simplification, solve the equations, and indicate which transformation and simplification you used. Show each step separately. Example: 4y 00 4y y 25 Transformations/Simplifications Used Divide both sides by 4 Simplify fractions. x 25 0 Transformations/Simplifications Used m 37 Transformations/Simplifications Used w 3. Transformations/Simplifications Used Chapter Linear Functions, Equations, and Inequalities

13 Problem 2 In an oil storage tank farm, one tank that is currently full contains 20,000 gallons of home heating oil. This tank will be emptied through a drain pipe at the rate of twenty five gallons per minute. In Problem 2 from the previous lesson, you defined variables to represent the quantities that were changing and used these variables to write an equation that showed the relationship between the quantities. You then used the equation to answer a number of questions about the situation. Use the equation 20,000 25t g to determine how long it will take for the tank to contain 5,000 gallons of heating oil. In this case, to solve the equation, you had to perform two transformations and two simplifications. In each of the following equations, perform the transformations and simplifications, solve the equations, and indicate which transformations and simplifications you used. Make sure to show each step separately.. 2w 8 4 Transformations/Simplifications Used 2. 2x 3 0 Transformations/Simplifications Used Lesson.2 Solving Linear Equations and Linear Inequalities in One Variable 3

14 m 37 Transformations/Simplifications Used f Transformations/Simplifications Used (3k 4) 2 Transformations/Simplifications Used 4 Chapter Linear Functions, Equations, and Inequalities

15 6. 2(4w) 6 5 Transformations/Simplifications Used 7..2s ( 9).5 Transformations/Simplifications Used 8. 3d 4d 2 5d Transformations/Simplifications Used Lesson.2 Solving Linear Equations and Linear Inequalities in One Variable 5

16 9. 3x 5 7x 4 Transformations/Simplifications Used 3x 0. 7 Transformations/Simplifications Used 4 6 Chapter Linear Functions, Equations, and Inequalities

17 Problem 3 In the first Tanks a Lot problem, you were asked: For what interval of time will there be less than 000 gallons of oil in the tank? This actually produces a linear inequality rather than an equation. The five different inequality symbols that are used in linear inequalities are listed. Next to each symbol, write its meaning. Take Note There is no difference in using transformations and simplifications with inequalities except when you multiply or divide both sides by a negative number. If x 5, then x is any number larger than 5, but if we multiply or divide both sides of this inequality by a negative number, for example 3, then we get 3x 5. If we let x 6, we have 3(6) 5 but 8 is not larger than 5. So when we multiply or divide by a negative, we must also reverse the direction of the inequality sign. For each of the following inequalities, perform the transformations and simplifications, solve the inequalities, and indicate which transformations and simplifications you used. Make sure to show each step separately. The answers to most inequalities are an infinite set of numbers, so we often graph these sets on a number line by using closed or open endpoints with shading of the line. Example: 2x 7 9 2x x 6 2x x 8 Transformations/Simplifications Used Add 7 to both sides Add/Subtract terms Divide both sides by 2 Simplify fractions Lesson.2 Solving Linear Equations and Linear Inequalities in One Variable 7

18 . 3c 8 3 Transformations/Simplifications Used 2. 2x 6 20 Transformations/Simplifications Used m 47 Transformations/Simplifications Used 8 Chapter Linear Functions, Equations, and Inequalities

19 h Transformations/Simplifications Used (3r 4) 2 Transformations/Simplifications Used Be prepared to share your work with another pair, group, or the entire class. Lesson.2 Solving Linear Equations and Linear Inequalities in One Variable 9

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21 .3 Running a 0K Slope Intercept Form of Linear Functions Objectives In this lesson, you will Define and calculate the slope of a linear function. Define and calculate the y-intercept of a linear function. Write linear equations in slope intercept form. Use the formula to calculate the slope of a linear equation. Graph linear functions using the slope and y-intercept. Key Terms slope y-intercept slope-intercept form Problem A runner is participating in a 0-kilometer road race and she is currently at the halfway point. She looks at her watch and notices that it has taken her exactly twenty five minutes to get to this point. Assume that she has run at a constant rate and will complete the race at this same rate.. Calculate the rate she is running by first solving for the amount of time it will take her to run km. Explain how you determined her rate. 2. At which kilometer mark is she currently? Lesson.3 Slope Intercept Form of Linear Functions 2

22 3. If she continues at this rate, how much of the race will she have completed in 0 more minutes? In 2 more minutes? 4. If she has currently been running for 25 minutes, how much of the race did she complete 5 minutes ago? 5. How many minutes would it take her to complete 8 kilometers of the race? Three fourths of the race? 6. What are the quantities that are changing after she has reached the 5 km mark? a. Independent quantity? b. Dependent quantity? 7. Assign a variable to each of the quantities, and write an equation that shows the relationship between these variables. 22 Chapter Linear Functions, Equations, and Inequalities

23 8. Use the answers to Questions through 7 to complete the following table, making sure to identify the quantities that are changing, the units that are used to measure these quantities, and the expressions representing each of these quantities. Then construct a graph. Quantity Name Unit Expression 9. Use your graph to estimate how much of the race will be left after she has run 20 more minutes. 0. For each minute she runs, how far does she go? This is her unit rate of change. In a linear function, this unit rate of change is called the slope. Lesson.3 Slope Intercept Form of Linear Functions 23

24 . Where is she currently? Describe the location of this point on the graph. What do we call this point on the graph? 2. In the equation you wrote for this situation in Question 7, are the slope and y-intercept obvious? Explain. Problem 2 Linear functions are often written in the form y mx b where m is the and b is the. This form is called the slope intercept form of a linear function. The slope is the unit rate of change. Each time the independent variable changes by one unit, the dependent variable increases or decreases by the value of the slope. The slope can be found by calculating the change in the dependent variable divided by the change in the independent variable. m change in dependent quantity change in independent quantity This is often written in a shorthand form as the following formula where is read as the change in, (x, y ) and (x 2, y 2 ), two ordered pairs or data points from the function, where y is the dependent variable, and x is the independent variable. m y x y 2 y x 2 x Calculate the slope and y-intercept for each of the following linear functions:. y 5x A linear function that passes through the points (0, 5) and (2, 5) 24 Chapter Linear Functions, Equations, and Inequalities

25 3. A linear function that passes through the points (2, 4) and ( 2, 7) 4. A linear function that passes through the points (8, 5) and ( 6, 9) If the linear function is in the slope intercept form, it enables you to graph the function quickly and easily using the y-intercept as an initial-value starting point and the slope as a unit rate of change. 5. For instance, in the equation y 3x 5, identify the slope and y-intercept. Lesson.3 Slope Intercept Form of Linear Functions 25

26 6. Using the equation y 3x 5, plot the y-intercept on the graph. Then plot the next point by moving one unit to the right and then moving up, if the slope is positive, or down, if the slope is negative, by the value of the slope. From this second point, repeat the process to plot as many points as you wish, and then draw your line through these points. For each of the following linear functions in slope intercept form, identify the slope and the y-intercept, and then construct their graphs using the slope and y-intercept. 7. y 2x 4 Slope y-intercept 26 Chapter Linear Functions, Equations, and Inequalities

27 8. y 4x Slope y-intercept 9. y 3x 5 Slope y-intercept Lesson.3 Slope Intercept Form of Linear Functions 27

28 0. y 2 Slope y-intercept 3 x 2. y 6 Slope y-intercept Be prepared to share your work with another pair, group, or the entire class. 28 Chapter Linear Functions, Equations, and Inequalities

29 .4 Pump It Up Standard Form of Linear Functions Objectives In this lesson, you will Write linear equations in standard form. Graph linear functions in standard form using intercepts. Transform linear equations in standard form to slope intercept form. Key Term standard form of a linear equation Problem After a water main break, a large building s basement was flooded to the ceiling. The local fire department provided two pump trucks to pump the water out of the basement. The first truck, No. 75, can pump 25 cubic feet of water per minute, and the second truck, No. 236, can pump 32 cubic feet of water per minute. The building s basement is a large rectangular prism, measuring 50 feet long, 20 feet wide, and 0 feet deep.. How many cubic feet of water are there in the basement? Explain how you determined this answer. 2. If both pump trucks are used for different amounts of time to pump out the basement, define variables for the time that each truck pumps, and then write an equation that represents this situation. Lesson.4 Standard Form of Linear Functions 29

30 3. If Truck No. 75 works for an hour, how many minutes must the other truck work to completely empty the basement? 4. If Truck No. 236 works for 90 minutes, how many minutes must the other truck work to completely empty the basement? 5. If only Truck No. 75 is used, how long would it take to empty the basement? Truck No. 236? 6. If Truck No. 236 empties 20,000 cubic feet of water, how many minutes must the other truck work to completely empty the basement? 30 Chapter Linear Functions, Equations, and Inequalities

31 7. If Truck No. 75 empties 20,000 cubic feet of water, how many minutes must the other truck work to completely empty the basement? 8. If Truck No. 75 works for three and one half hours, how many cubic feet of water are left for the other truck to pump? 9. Complete the following table and graph the times for each truck. Quantity Name Unit Expression 0. What kind of function does this situation produce? Explain how you know. Lesson.4 Standard Form of Linear Functions 3

32 . Using two data points/ordered pairs from your table, calculate the slope. What is the y-intercept? 2. Rewrite this equation in slope intercept form. Problem 2 A linear function of the form Ax By C where A, B, and C are integer constants with no common factor larger than is said to be the standard form of a linear equation. One advantage of the standard form is that it enables you to calculate both the x- and y-intercepts easily. For each of the following linear equations written in standard form, calculate both the x- and y-intercepts, and then use these intercepts to graph the function.. 4x 9y 44 x-intercept y-intercept 32 Chapter Linear Functions, Equations, and Inequalities

33 2. 5x 7y 35 x-intercept y-intercept 3. 8x 7y 56 x-intercept y-intercept Lesson.4 Standard Form of Linear Functions 33

34 When a linear equation is written in standard form, it can be transformed into the slope intercept form by solving for y. For each of the following linear equations in standard form, transform it into slope intercept form, calculate the slope and y-intercept, and construct its graph. 4. 3x 6y 5 slope y-intercept 34 Chapter Linear Functions, Equations, and Inequalities

35 5. 5x y slope y-intercept 6. 8x 7y 56 slope y-intercept Lesson.4 Standard Form of Linear Functions 35

36 Sometimes you may be asked to transform a linear function that is in slope intercept form into standard form. For each of the following linear functions in slope intercept form, transform it into standard form. 7. y 3x 4 8. y 2 3 x 6 9. y 3.4x y 7 4 x 9 8 Be prepared to share your work with another pair, group, or the entire class. 36 Chapter Linear Functions, Equations, and Inequalities

37 .5 Shifts and Flips Basic Functions and Linear Transformations Objectives In this lesson, you will Define basic functions. Use translations, dilations, and reflections to transform linear functions. Graph parallel lines. Graph perpendicular lines. Key Terms basic function dilation reflection line of reflection Problem We have now worked with two different forms of linear functions, slope intercept form and standard form. You should remember that a function is defined as a relation for which every input value has one and only one output value. We are going to look at linear functions from the view of a family of functions. The most basic form for a linear function is y x Remember A dilation is a transformation of a figure in which the figure stretches or shrinks with respect to a fixed point. Remember A reflection is a transformation in which a figure is reflected, or flipped, in a given line called the line of reflection. which is called the basic function. Any linear function can be constructed through a series of transformations to the basic function. Algebraic Transformations Add a constant Subtract a constant Multiply or divide by a positive constant Multiply by Graphical Transformations Shift up Shift down Dilation Reflection Use the table to graph the functions, and indicate the transformations, both in terms of transforming the equation and the graph, which were performed on the basic function to arrive at the transformed function. Lesson.5 Basic Functions and Linear Transformations 37

38 . Basic function y x Algebraic transformation: Graphical transformation: 2. y x 3 Algebraic transformation: Graphical transformation: 38 Chapter Linear Functions, Equations, and Inequalities

39 3. y x 4 Algebraic transformation: Graphical transformation: 4. y 2x Algebraic transformation: Graphical transformation: Lesson.5 Basic Functions and Linear Transformations 39

40 5. y 2x Algebraic transformation: Graphical transformation: 6. y 3x Algebraic transformation: Graphical transformation: 40 Chapter Linear Functions, Equations, and Inequalities

41 7. y 4x Algebraic transformation: Graphical transformation: 8. y 3x 5 Algebraic transformation: Graphical transformation: Lesson.5 Basic Functions and Linear Transformations 4

42 9. y 2 3 x Algebraic transformation: Graphical transformation: 0. y 2 x 3 Algebraic transformation: Graphical transformation: 42 Chapter Linear Functions, Equations, and Inequalities

43 For each of the following equations of linear functions, describe the transformations you would need to perform to the graph of the basic function in order to transform it into the given function.. y 4x 2. y x 7 3. y 2x 7 4. y 7x Problem 2 Graph the following equations on the same grid.. y 2x and y 2x 5 2. Describe how the graphs are related geometrically. Lesson.5 Basic Functions and Linear Transformations 43

44 3. Graph the following equations on the same grid. y 3x and y 3x 5 4. Describe how the graphs are related geometrically. 5. What conclusion might you make about equations with the same slope? 6. The graphs of y 2x and y are shown on the graph. 2 x y 8 6 y = 2x y = 2 x x From the x-axis, draw a line segment vertically from (2, 0) to the line y 2x to form a right triangle. From the x-axis, draw a line segment vertically from ( 4, 0) to the line y to form a second right triangle. 2 x 44 Chapter Linear Functions, Equations, and Inequalities

45 7. Using what you know from geometry, why are the two triangles congruent? What can you conclude about the angles formed by the intersecting lines y 2x and y? 2 x 8. The graphs of y 3x and y are on the grid. 3 x y 8 6 y = 3x y = 3 x x 6 8 From the x-axis, draw a line segment vertically from (2, 0) to the line y 3x to form a right triangle. From the x-axis, draw a line segment vertically from ( 4, 0) to the line y to form a second right triangle. 3 x 9. Using what you know from geometry, why are the two right triangles congruent? What can you conclude about the angles formed by the intersecting lines y 3x and y? 3 x 0. What conclusion can you draw about linear functions with related slopes? Be prepared to share your work with another pair, group, or the entire class. Lesson.5 Basic Functions and Linear Transformations 45

46 46 Chapter Linear Functions, Equations, and Inequalities

47 .6 Inventory and Sand Determining the Equations of Linear Functions Objectives In this lesson, you will Determine the equation of linear functions when given the slope and the y-intercept. given the slope and one point on the line. given two points on the line. given the equation of a line parallel to the line and a point on the line. given the equation of a line perpendicular to the line and a point on the line. Key Terms point-slope form two-point form parallel lines perpendicular lines Problem A company discovers that the person who was to keep track of the inventory of a vital component left without leaving any records. Assume that the company uses the same number of these components every day, and they had 724 on hand on the 0th of the month. Three days later they have 688 on hand.. How many components do they use per day on average? 2. How many components did they have at the beginning of the month? 3. Define variables for both the independent and dependent variables, and write a linear function that represents the number of components on hand based on the 0th day of the month. Lesson.6 Determining the Equations of Linear Functions 47

48 4. Use the function to determine how long before the inventory is gone. Problem 2 A company that sells river sand starts the week with 5,000 tons of sand, and they sell about 5 tons of sand per day.. Define variables for both the independent and dependent variables, and write a linear function that represents the amount of sand on hand based on the day. 2. Use the function to calculate how long before there are only 8000 tons of sand. 48 Chapter Linear Functions, Equations, and Inequalities

49 Problem 3 A gasoline station sells approximately 250 gallons of gas per day. Assume that they sell the same amount per day and they receive a shipment at the beginning of each month. On the fifth day after a delivery, they have 26,450 gallons on hand.. How many gallons of gas do they sell per day on average? 2. How many gallons of gas did they have at the beginning of the month? 3. Define variables for both the independent and dependent variables, and write a linear function that represents the amount of gas on hand based on the number of days since the delivery. 4. Use the function to calculate how long before the amount of gas will reach 5000 gallons. In each case, you defined a linear function in order to model a situation and then used the function to answer important questions. The ability to write a linear function to model a situation is very useful. In Problem, you were given two data points or ordered pairs. In Problem 2, you were given an initial value (y-intercept) and the unit rate of change (slope). In Problem 3, you were given one data point and the unit rate of change. The following is a list of different situations that you may be presented with for which you will need to write the equation of a linear function. A. Given the slope and the y-intercept (Problem 2) B. Given the slope and one point on the line that is not the y-intercept (Problem 3) C. Given two points on the line (Problem ) D. Given a point that the line passes through and the equation of a parallel line E. Given a point that the line passes through and the equation of a perpendicular line Lesson.6 Determining the Equations of Linear Functions 49

50 For each of these situations, there are several ways to determine the equations of the linear functions. A. Given the slope and the y-intercept (Problem 2) The equation can be written directly using the slope intercept form y mx b Example: Slope 4 and y-intercept 5 y 4x 5 B. Given the slope and one point on the line that is not the y-intercept (Problem 3) You can determine the equation using the slope intercept form by substituting in the value of m and then substituting the x and y coordinates from the ordered pair in y mx b and solving for b. Example: Slope 2 and the line passes through the point (3, 4) y 2x b substitute for m 4 2(3) b substitute in x and y coordinates 0 b y 2x 0 You can use another form called the point slope form, ( y y ) m(x x ) where m is the slope and (x, y ) is a point on the line. Example: Slope 2 and the line passes through the point (3, 4) ( y 4) 2(x 3) y 4 2x 6 y 2x 0 C. Given two points on the line (Problem ) You can determine the equation using the slope intercept form by calculating the slope by using the slope formula m y and then substituting the x x y 2 y x 2 x and y coordinates from one of the ordered pairs in y mx b and solving for b. Example: Determine the equation of the line that passes through (2, 4) and ( 3, 4). m y x y 2 y x 2 x y 2x b 4 2(2) b 8 b y 2x 8 50 Chapter Linear Functions, Equations, and Inequalities

51 You can use another form called the two point form: y 2 y y y x 2 x x x Example: Determine the equation of the line that passes through (2, 4) and ( 3, 4). 4 4 y x 2 D. Given a point that the line passes through and the equation of a parallel line You can determine the equation of the line by identifying the slope of the given line. Parallel lines must have equal slopes. Then calculate the y-intercept by substituting the x and y coordinates from the ordered pair in y mx b and solving for b. Example: Determine the equation of a line parallel to the line y 4x 7 and passing through the point ( 2, 3). The slope of the given line is 4. y 4x b 5 b y 4x 5 0 y 4 5 x 2 2(x 2) y 4 2x 4 y 4 3 4( 2) b y 2x 8 E. Given a point that the line passes through and the equation of a perpendicular line You can determine the equation of the line by identifying the slope of the given line. Perpendicular lines have slopes that are negative reciprocals. Then calculate the y-intercept by substituting the x and y coordinates from the ordered pair in y mx b and solving for b. Example: Determine the equation of the line perpendicular to the line y 3x 6 and passing through the point ( 4, 5). Lesson.6 Determining the Equations of Linear Functions 5

52 The slope of the given line is 3, and the slope of the perpendicular line is. 3 y 3 x b 5 ( 4) b 3 3 b y 3 x 3 For each of the following situations, determine the equation of the line.. The line has a slope of 4 that passes through the point (4, 0). 2. The line passes through the points (3, 7) and ( 5, 9). 3. The line is parallel to the line 2x 4y 9 and passes through the point (, ). 52 Chapter Linear Functions, Equations, and Inequalities

53 4. The line is perpendicular to the line y and passes through the point 5 x 6 5 (0, 2). 5. The line has a slope of 2 and y-intercept of The line is perpendicular to the line y x 7 and passes through the point (0, 5). 7. The line has a slope of 9 and a y-intercept of The line passes through the points ( 9, 5) and ( 2, 4). Lesson.6 Determining the Equations of Linear Functions 53

54 9. The line is parallel to the line y x 9 and passes through the point ( 9, 5). 0. The line has a slope of 2. and passes through the point ( 3, 7). Be prepared to share your work with another pair, group, or the entire class. 54 Chapter Linear Functions, Equations, and Inequalities

55 .7 Absolutely! Absolute Value Equations and Inequalities Objectives In this lesson, you will Write absolute value equations and inequalities in one and two variables. Solve absolute value equations and inequalities in one and two variables. Graph absolute value equations and inequalities in one and two variables. Key Terms absolute value absolute value equation absolute value inequality compound inequality Problem Solving Absolute Value Equations Absolute value is a concept that has wide application in higher mathematics, and you may remember that the absolute value of a number is the distance from the number to zero on the number line. Distance is always positive. The more technical definition is x x if x 0 x x if x 0. Calculate the absolute value of each of the following: a. 7 b. 7 c. 0 d e. 2. Calculate the value(s) for each of the following variables that have the given absolute values: a. a 9 b. c 3.4 c. y 0 d. d 5 Lesson.7 Absolute Value Equations and Inequalities 55

56 When we solve a linear equation with one variable, we can perform the various transformations and simplifications to isolate the variable and calculate our solution. An absolute value equation contains an absolute value and may have more than one solution. For instance: x 5 8 x 5 8 or x 5 8 x or x x 3 or x 3 Add 5 to both sides Combine like terms or x 5 8 x 5 8 x x 3 or x 3 3. Solve each of the following equations: Add 5 to both sides Combine like terms a. x 5 0 b. 2x 5 7 c x Chapter Linear Functions, Equations, and Inequalities

57 d. 5x 2 8 e. 7x What do you notice about the solution(s) to the equations in Question 3? How do the solutions differ from those of other equations with one variable that you have solved before? Problem 2 Graphing Absolute Value Functions When we solve linear equations in two variables, the solution is a set of ordered pairs that satisfies the equation.. Graph the following equation on the grid: y x 2 a. For what values of x are the values of y 0? Lesson.7 Absolute Value Equations and Inequalities 57

58 b. Graph y x 2 on the same grid by plotting the points in the table below: x y c. What do you notice about this graph? d. What is the least value of y that is a solution of y x 2? For what value of x does this occur? What is this ordered pair called? 2. Graph the solutions for each of the following equations: a. y x 3 58 Chapter Linear Functions, Equations, and Inequalities

59 b. y 2x 3 c. y x 3 Lesson.7 Absolute Value Equations and Inequalities 59

60 Problem 3 Solving Linear Inequalities with Absolute Value in One Variable Linear inequalities that contain absolute value, an absolute value inequality, may have infinite solutions, and the solutions are represented by segments or rays on the number line.. To solve the inequality x 2 4, first solve the equation. a. x 2 4 b. Graph these two points on the number line below. c. Will these endpoints be included in this inequality? How do you know? d. These two points divide the number line into three distinct portions; choose a number from each of the regions and substitute them into the inequality to determine which of these portions satisfy the inequality. Shade the portion(s) that satisfy the inequality. e. We can also solve the inequality algebraically by rewriting it as a compound inequality. x 2 4 x 2 4 and x 2 4 Why? Can be rewritten as 60 Chapter Linear Functions, Equations, and Inequalities

61 f. Solve the compound inequality. How does this solution correspond to the solution you calculated in part (a)? When a solution is confined in a region, both the lower and upper bounds must be specified. 2. Solve the following inequality graphically and then algebraically. x 2 When a solution is confined outside a region, x is below the lower bound or above the upper bound of the region. Solve each of the following inequalities and graph their solutions on the number lines provided. 3. 2x x 3 7 Lesson.7 Absolute Value Equations and Inequalities 6

62 x x x Chapter Linear Functions, Equations, and Inequalities

63 Problem 4 Solving Linear Inequalities with Absolute Value in Two Variables. Graph the following equation on the grid: y x 3 a. Which portions of the coordinate plane has ordered pairs so that the values of y x 3? y x 3? y x 3? The solution set of a linear inequality with an absolute value is the portion of the coordinate plane that satisfies the inequality. To indicate the solution set, we shade the portion. If the line segments are included in the solution, we use a solid line, and if they are not included, we use a dotted or dashed line. b. Shade the solution set of y x 3 on the grid in Question. Lesson.7 Absolute Value Equations and Inequalities 63

64 2. Graph the solutions for each of the following inequalities: a. y x 2 b. y 3x 64 Chapter Linear Functions, Equations, and Inequalities

65 c. y 4x 8 Be prepared to share your work with another pair, group, or the entire class. Lesson.7 Absolute Value Equations and Inequalities 65

66 66 Chapter Linear Functions, Equations, and Inequalities

67 .8 Inverses and Pieces Functional Notation, Inverses, and Piecewise Functions Objectives In this lesson, you will Write linear functions using functional notation. Write compositions of functions. Determine inverses of linear functions. Define piecewise linear functions. Graph piecewise linear functions. Key Terms relation domain range function inverse operation functional notation identity function inverse function composition of functions piecewise functions Problem A relation is any correspondence between a set of input values and output values. The set of all inputs is the domain of the relation. The set of all outputs is the range of the relation. A function is a special type of relation in which for every member of its domain is associated with exactly one member of its range. In other words, a function is a relationship in which each input value has a unique output value. For example, If you are selling candy bars for a fundraiser, the relationship between the number of candy bars sold and the resulting income is a function. For any amount of candy bars sold (the input or domain), there is a unique value for the income (the output or range). If you are counting how much money you have based on the total number of coins in your pocket, the relationship is not a function. The amount of money may vary depending on the types of coins you have. While this is not an example of a function, it is still a relation.. Every holder of a social security card in the United States is assigned a nine-digit social security number. a. Let the domain be all assigned social security numbers and the range be the names of all social security cardholders. Is this relationship a function? Why or why not? Lesson.8 Functional Notation, Inverses, and Piecewise Functions 67

68 b. Consider reversing the situation. Let the domain be the names of all social security cardholders and the range be all assigned social security numbers. Is this reverse relationship a function? Why or why not? 2. Every member on the school football team is assigned a number. a. Let the domain be all the football players on the school team and the range be all of the assigned player numbers. Is this relationship a function? Why or why not? b. Reverse the situation. Let the domain be all assigned player numbers and the range be all of the football players on the school team. Is this reverse relationship a function? Why or why not? 3. Each person has a favorite color. a. Let the domain be all people in the world and let the range be all the colors. Is this relationship a function? Why or why not? b. Reverse the situation so that the domain is all of the colors and the range is all people in the world. Is the reverse relationship a function? Why or why not? Problem 2 In Questions through 3, you considered general relationships and their reverse relationships and decided whether or not they were functions. Now you will look at more specific situations and determine their reverse. In other words, you will determine how to undo the situation. Undoing, working backwards, or retracing steps to return to an original value or position is referred to as the inverse operation. Write a phrase, expression, or sentence for the inverse of each given action.. Open a door. 2. Turn on a light. 68 Chapter Linear Functions, Equations, and Inequalities

69 3. Add 6 to a number. 4. Walk 2 blocks east and then 3 blocks south. 5. Get into the deep end of a pool and swim to the shallow end. 6. Multiply a number by 3 and subtract For Question 6, Jan gave the inverse as Divide by 3 and then add 5. Marcus gave the inverse as Add 5 and then divide by 3. Who is correct? Why? Problem 3 In this chapter, you have been working with linear relations in slope intercept form, y mx b, and standard form, Ax By C. Remember Functional notation is often used to represent functions. f(x) is read f of x or the value of the function f at x. For example, if f(x) 3x 5 then f( 2) 3( 2) 5. b. g(x) 7x 8, calculate g(0), g(5). Are the linear relations functions? Why or why not? 2. For each of the following functions written in functional notation, calculate its value for the given values of the independent variable. a. f(x) 3x, for x 2, 7 c. h(x) x 2 8x, calculate h( ), h(3) d. f(x) x, calculate f( ), f(3) In Question 2d, the function f assigns every value of x to itself; this is called the identity function. Lesson.8 Functional Notation, Inverses, and Piecewise Functions 69

70 Take Note Applying one function to the answer of another function is called the composition of functions. f g(x) or f(g(x)) is read f of g of x or the value of the function f at the value of the function g at x. For example, if f(x) 3x 5 and g(x) 2x then f(g( 2)) f(2( 2)) f( 4) 3( 4) To determine the inverse of a linear function, you need to derive the function that undoes the original function. For each of the following functions determine its inverse function. a. f( x) 3x b. g( x) x 8 c. h( x) x.2 4. For each of the functions and inverses from Question 3, calculate each of the following: a. f ( f(3)) f ( f( 2)) f (x) g (x) h ( x) b. g ( g( )) g (g(6)) g ( g(x)) c. h (h(6)) f ( f(x)) h (h( 9.6)) h (h(x)) 5. In each case, what happens when an inverse of a function is composed with the function? A function f is the set of all ordered pairs (x, y) or (x, f(x)), where for every value of x there is one and only one value of y or f(x). To represent any linear equation in two variables using functional notation, first solve for y and replace y with f(x). The inverse of this function f ( x) is the set of all ordered pairs (y, x) or (f(x), x). To derive the inverse function, solve for the dependent variable and then reverse the variables. For each of the following, first write the linear equation in two variables in functional notation and then determine its inverse. 6. y 4x 8 f( x) f (x) 70 Chapter Linear Functions, Equations, and Inequalities

71 7. 3x 7y 9 f(x) f (x) 8. 4x 2y 0 f(x) f ( x) 9. 3y 7x f(x) f ( x) Problem 4 You have been offered a new job selling medical supplies. As part of your compensation package, you are to receive a commission on your total sales each year with a rate that changes as you sell more. You will receive % of your total sales for the first $20,000 sold 2% of your total sales from over $20,000 to $40,000 sold 5% of your total sales over $40,000. Define variables for your total sales and your commission. a. Use these variables to write a function for your total commissions up to $20,000. b. What would be your commission if you sold $5000 of medical supplies? $0,000? $20,000? c. Use these variables to write a function for your total commissions from over $20,000 to $40,000. d. What would be your commission if you sold $25,000 of medical supplies? $30,000? $40,000? e. Use these variables to write a function for your total commissions over $40,000. Lesson.8 Functional Notation, Inverses, and Piecewise Functions 7

72 f. What would be your commission if you sold $45,000 of medical supplies? $50,000? $60,000? g. Use the values to complete the table, making sure to find the three different expressions for the different commissions. Construct a graph of your commission with total sales from $0 to $60,000 using the values from parts (b), (d), and (f) on the grid. Using what you know about linear graphs, connect these points. Quantity Name Unit Expression 72 Chapter Linear Functions, Equations, and Inequalities

73 2. Using this graph, answer the following questions: a. How is this graph different from the others you have constructed? b. Is this graph a function? Why or why not? c. A relation like this is called a piecewise relation. Why? Piecewise functions are functions which are defined with different functional relationships between the independent and dependent variables over different domains. This function could be written as a single function as f(x) 0.0x x 20, (x 20,000) 20,000 x 40, (x 40,000) x 40, Graph each of the following piecewise functions. 2x x 4 a. f(x) 2x 7 x 4 Lesson.8 Functional Notation, Inverses, and Piecewise Functions 73

74 x x 0 b. f(x) 3x 0 x 4 x x 4 Be prepared to share your work with another pair, group, or the entire class. 74 Chapter Linear Functions, Equations, and Inequalities