Linear Functions, Equations, and Inequalities


 Mariah Cameron
 4 years ago
 Views:
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 onestep and twostep 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
20 20 Chapter Linear Functions, Equations, and Inequalities
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 yintercept 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 yintercept. Key Terms slope yintercept slopeintercept form Problem A runner is participating in a 0kilometer 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 yintercept 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 yintercept 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 yintercept as an initialvalue starting point and the slope as a unit rate of change. 5. For instance, in the equation y 3x 5, identify the slope and yintercept. Lesson.3 Slope Intercept Form of Linear Functions 25
26 6. Using the equation y 3x 5, plot the yintercept 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 yintercept, and then construct their graphs using the slope and yintercept. 7. y 2x 4 Slope yintercept 26 Chapter Linear Functions, Equations, and Inequalities
27 8. y 4x Slope yintercept 9. y 3x 5 Slope yintercept Lesson.3 Slope Intercept Form of Linear Functions 27
28 0. y 2 Slope yintercept 3 x 2. y 6 Slope yintercept 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 yintercept? 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 yintercepts easily. For each of the following linear equations written in standard form, calculate both the x and yintercepts, and then use these intercepts to graph the function.. 4x 9y 44 xintercept yintercept 32 Chapter Linear Functions, Equations, and Inequalities
33 2. 5x 7y 35 xintercept yintercept 3. 8x 7y 56 xintercept yintercept 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 yintercept, and construct its graph. 4. 3x 6y 5 slope yintercept 34 Chapter Linear Functions, Equations, and Inequalities
35 5. 5x y slope yintercept 6. 8x 7y 56 slope yintercept 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 xaxis, draw a line segment vertically from (2, 0) to the line y 2x to form a right triangle. From the xaxis, 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 xaxis, draw a line segment vertically from (2, 0) to the line y 3x to form a right triangle. From the xaxis, 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 yintercept. 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 pointslope form twopoint 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 (yintercept) 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 yintercept (Problem 2) B. Given the slope and one point on the line that is not the yintercept (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 yintercept (Problem 2) The equation can be written directly using the slope intercept form y mx b Example: Slope 4 and yintercept 5 y 4x 5 B. Given the slope and one point on the line that is not the yintercept (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 yintercept 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 yintercept 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 yintercept 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 yintercept 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 ninedigit 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
16.3. In times before global positioning systems (GPS) or navigation devices, hikers would. The Inverse Undoes What a Function Does
The Inverse Undoes What a Function Does Inverses of Linear Functions.3 Learning Goals In this lesson, you will: Determine the inverse of a given situation using words. Determine the inverse of a function
More informationLooking Ahead to Chapter 4
Looking Ahead to Chapter Focus In Chapter, you will learn about functions and function notation, and you will find the domain and range of a function. You will also learn about real numbers and their properties,
More informationSkills Practice Skills Practice for Lesson 1.1
Skills Practice Skills Practice for Lesson. Name Date Tanks a Lot Introduction to Linear Functions Vocabulary Define each term in your own words.. function 2. linear function 3. independent variable 4.
More informationMatrix Operations and Equations
C H A P T ER Matrix Operations and Equations 200 Carnegie Learning, Inc. Shoe stores stock various sizes and widths of each style to accommodate buyers with different shaped feet. You will use matrix operations
More informationExponential Functions
CONDENSED LESSON 5.1 Exponential Functions In this lesson, you Write a recursive formula to model radioactive decay Find an exponential function that passes through the points of a geometric sequence Learn
More informationMath 3 Variable Manipulation Part 7 Absolute Value & Inequalities
Math 3 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,
More informationMidterm: Wednesday, January 23 rd at 8AM Midterm Review
Name: Algebra 1 CC Period: Midterm: Wednesday, January 23 rd at 8AM Midterm Review Unit 1: Building Blocks of Algebra Number Properties (Distributive, Commutative, Associative, Additive, Multiplicative)
More informationLearning Target #1: I am learning to compare tables, equations, and graphs to model and solve linear & nonlinear situations.
8 th Grade Honors Name: Chapter 2 Examples of Rigor Learning Target #: I am learning to compare tables, equations, and graphs to model and solve linear & nonlinear situations. Success Criteria I know I
More informationChapter Two B: Linear Expressions, Equations, and Inequalities
Chapter Two B: Linear Expressions, Equations, and Inequalities Index: A: Intro to Inequalities (U2L8) Page 1 B: Solving Linear Inequalities (U2L9) Page 7 C: Compound Inequalities (And) (U2L10/11) Page
More informationChapter Four: Linear Expressions, Equations, and Inequalities
Chapter Four: Linear Expressions, Equations, and Inequalities Index: A: Intro to Inequalities (U2L8) B: Solving Linear Inequalities (U2L9) C: Compound Inequalities (And) (U2L10/11) D: Compound Inequalities
More informationCourse Readiness and Skills Review Handbook (Topics 110, 17) (240 topics, due. on 09/11/2015) Course Readiness (55 topics)
Course Name: Gr. 8 Fall 2015 Course Code: C6HNHTEK9E ALEKS Course: Middle School Math Course 3 Instructor: Mr. Fernando Course Dates: Begin: 08/31/2015 End: 06/17/2016 Course Content: 642 Topics (637
More informationNOTES. [Type the document subtitle] Math 0310
NOTES [Type the document subtitle] Math 010 Cartesian Coordinate System We use a rectangular coordinate system to help us map out relations. The coordinate grid has a horizontal axis and a vertical axis.
More informationMATH 1113 Exam 1 Review
MATH 1113 Exam 1 Review Topics Covered Section 1.1: Rectangular Coordinate System Section 1.3: Functions and Relations Section 1.4: Linear Equations in Two Variables and Linear Functions Section 1.5: Applications
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part II 1 st Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part II 1 st Nine Weeks, 20162017 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
More informationUnit 5 Algebraic Investigations: Quadratics and More, Part 1
Accelerated Mathematics I Frameworks Student Edition Unit 5 Algebraic Investigations: Quadratics and More, Part 1 2 nd Edition March, 2011 Table of Contents INTRODUCTION:... 3 Notes on Tiling Pools Learning
More informationCLASS NOTES: 2 1 thru 2 3 and 1 1 Solving Inequalities and Graphing
page 1 of 19 CLASS NOTES: 2 1 thru 2 3 and 1 1 Solving Inequalities and Graphing 1 1: Real Numbers and Their Graphs Graph each of the following sets. Positive Integers: { 1, 2, 3, 4, } Origin: { 0} Negative
More informationName Class Date. What is the solution to the system? Solve by graphing. Check. x + y = 4. You have a second point (4, 0), which is the xintercept.
61 Reteaching Graphing is useful for solving a system of equations. Graph both equations and look for a point of intersection, which is the solution of that system. If there is no point of intersection,
More informationSection 3.4 Writing the Equation of a Line
Chapter Linear Equations and Functions Section.4 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
More informationArchdiocese of Washington Catholic Schools Academic Standards Mathematics
8 th GRADE Archdiocese of Washington Catholic Schools Standard 1  Number Sense Students know the properties of rational* and irrational* numbers expressed in a variety of forms. They understand and use
More informationAlgebra 1 Spencer Unit 4 Notes: Inequalities and Graphing Linear Equations. Unit Calendar
Algebra 1 Spencer Unit 4 Notes: Inequalities and Graphing Linear Equations Unit Calendar Date Topic Homework Nov 5 (A ) 6.1 Solving Linear Inequalities +/ 6.2 Solving Linear Inequalities x/ 6.3 Solving
More informationChapter 12 Add and Subtract Integers
Chapter 12 Add and Subtract Integers Absolute Value of a number is its distance from zero on the number line. 5 = 5 and 5 = 5 Adding Numbers with the Same Sign: Add the absolute values and use the sign
More informationPreAP Algebra I Problems for the First Semester Exam
This is not a semester exam, but problems that you could use on a semester exam that are similar to some of the problems from the unit quizzes 1. Stephanie left home at 8:30 and rode her bicycle at a steady
More informationDefine the word inequality
Warm Up: Define the word inequality Agenda: Objective Students can solve linear inequalities in one variable, including equations with coefficients represented by letters. Define Inequalities One & Two
More informationFoundations of High School Math
Foundations of High School Math This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to
More informationAlgebra I Practice Exam
Algebra I This practice assessment represents selected TEKS student expectations for each reporting category. These questions do not represent all the student expectations eligible for assessment. Copyright
More informationAlgebra 1 EndofCourse Assessment Practice Test with Solutions
Algebra 1 EndofCourse Assessment Practice Test with Solutions For Multiple Choice Items, circle the correct response. For Fillin Response Items, write your answer in the box provided, placing one digit
More informationEquations and Inequalities in One Variable
Name Date lass Equations and Inequalities in One Variable. Which of the following is ( r ) 5 + + s evaluated for r = 8 and s =? A 3 B 50 58. Solve 3x 9= for x. A B 7 3. What is the best first step for
More informationAlgebra 1 S1 Lesson Summaries. Lesson Goal: Mastery 70% or higher
Algebra 1 S1 Lesson Summaries For every lesson, you need to: Read through the LESSON REVIEW which is located below or on the last page of the lesson and 3hole punch into your MATH BINDER. Read and work
More informationCalifornia 5 th Grade Standards / Excel Math Correlation by Lesson Number
(Activity) L1 L2 L3 Excel Math Objective Recognizing numbers less than a million given in words or place value; recognizing addition and subtraction fact families; subtracting 2 threedigit numbers with
More informationJune If you want, you may scan your assignment and convert it to a.pdf file and it to me.
Summer Assignment PreCalculus Honors June 2016 Dear Student: This assignment is a mandatory part of the PreCalculus Honors course. Students who do not complete the assignment will be placed in the regular
More informationMAT116 Final Review Session Chapter 1: Equations, Inequalities, and Modeling
MAT116 Final Review Session Chapter 1: Equations, Inequalities, and Modeling Solving Equations: Types of Equations in Chapter 1: Linear Absolute Value Quadratic Rational To SOLVE an equation means to find
More information(MATH 1203, 1204, 1204R)
College Algebra (MATH 1203, 1204, 1204R) Departmental Review Problems For all questions that ask for an approximate answer, round to two decimal places (unless otherwise specified). The most closely related
More informationExample #1: Write an Equation Given Slope and a Point Write an equation in slopeintercept form for the line that has a slope of through (5,  2).
Algebra II: 24 Writing Linear Equations Date: Forms of Equations Consider the following graph. The line passes through and. Notice that is the yintercept of. You can use these two points to find the
More informationUnit 3: Linear and Exponential Functions
Unit 3: Linear and Exponential Functions In Unit 3, students will learn function notation and develop the concepts of domain and range. They will discover that functions can be combined in ways similar
More informationRelations and Functions
Lesson 5.1 Objectives Identify the domain and range of a relation. Write a rule for a sequence of numbers. Determine if a relation is a function. Relations and Functions You can estimate the distance of
More informationUNIT 5 INEQUALITIES CCM6+/7+ Name: Math Teacher:
UNIT 5 INEQUALITIES 20152016 CCM6+/7+ Name: Math Teacher: Topic(s) Page(s) Unit 5 Vocabulary 2 Writing and Graphing Inequalities 3 8 Solving OneStep Inequalities 9 15 Solving MultiStep Inequalities
More informationName: Features of Functions 5.1H
Name: Features of Functions 5.1H Ready, Set, Go! Ready Topic: Solve systems by graphing For each system of linear equations: a. Graph the system b. Find the x and yintercepts for each equation c. Find
More informationCCGPS UNIT 1 Semester 1 COORDINATE ALGEBRA Page 1 of 33. Relationships Between Quantities Name:
CCGPS UNIT 1 Semester 1 COORDINATE ALGEBRA Page 1 of 33 Relationships Between Quantities Name: Date: Reason quantitatively and use units to solve problems. MCC912.N.Q.1 Use units as a way to understand
More informationGrade 6  SBA Claim 1 Example Stems
Grade 6  SBA Claim 1 Example Stems This document takes publicly available information about the Smarter Balanced Assessment (SBA) in Mathematics, namely the Claim 1 Item Specifications, and combines and
More informationVariables and Expressions
Variables and Expressions A variable is a letter that represents a value that can change. A constant is a value that does not change. A numerical expression contains only constants and operations. An algebraic
More informationSY1415 Algebra Exit Exam  PRACTICE Version
Student Name: Directions: Solve each problem. You have a total of 90 minutes. Choose the best answer and fill in your answer document accordingly. For questions requiring a written response, write your
More informationAlgebra Supplement Homework Packet #1
Algebra Supplement Homework Packet #1 Day 1: Fill in each blank with one of the words or phrases listed below. Distributive Real Reciprocals Absolute value Opposite Associative Inequality Commutative Whole
More informationMiddle School Math Course 2
Middle School Math Course 2 This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet
More informationNC Math 3 Modelling with Polynomials
NC Math 3 Modelling with Polynomials Introduction to Polynomials; Polynomial Graphs and Key Features Polynomial Vocabulary Review Expression: Equation: Terms: o Monomial, Binomial, Trinomial, Polynomial
More informationSection 2.1 Objective 1: Determine If a Number Is a Solution of an Equation Video Length 5:19. Definition A in is an equation that can be
Section 2.1 Video Guide Linear Equations: The Addition and Multiplication Properties of Equality Objectives: 1. Determine If a Number Is a Solution of an Equation 2. Use the Addition Property of Equality
More informationUnit 5. Linear equations and inequalities OUTLINE. Topic 13: Solving linear equations. Topic 14: Problem solving with slope triangles
Unit 5 Linear equations and inequalities In this unit, you will build your understanding of the connection between linear functions and linear equations and inequalities that can be used to represent and
More informationNC Math 1. Released Items. North Carolina EndofCourse Assessment. Published October 2018
Released Items Published October 2018 NC Math 1 North Carolina EndofCourse Assessment Public Schools of North Carolina Department of Public Instruction State Board of Education Division of Accountability
More informationPre Algebra. Curriculum (634 topics)
Pre Algebra This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs.
More informationSystems of Equations and Inequalities
1 Systems of Equations and Inequalities 2015 03 24 2 Table of Contents Solving Systems by Graphing Solving Systems by Substitution Solve Systems by Elimination Choosing your Strategy Solving Systems of
More informationPrep for College Algebra
Prep for College Algebra This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (219 topics + 85 additional topics)
More informationAlgebra Readiness. Curriculum (445 topics additional topics)
Algebra Readiness This course covers the topics shown below; new topics have been highlighted. Students navigate learning paths based on their level of readiness. Institutional users may customize the
More informationWriting and Solving Equations
Writing and Solving Equations Melody s Music Solution Lesson 61 Modeling and Writing TwoStep Equations ACTIVITY 6 Learning Targets: Use variables to represent quantities in realworld problems. Model
More informationGlossary. alternate exterior angles. absolute value function. Additive Identity. Additive Inverse. alternate interior angles.
Glossary A absolute value function An absolute value function is a function that can be written in the form f(x) x where x is any number. Additive Identity The number is the additive identity because when
More informationUNIT 2: REASONING WITH LINEAR EQUATIONS AND INEQUALITIES. Solving Equations and Inequalities in One Variable
UNIT 2: REASONING WITH LINEAR EQUATIONS AND INEQUALITIES This unit investigates linear equations and inequalities. Students create linear equations and inequalities and use them to solve problems. They
More informationSolving Equations Quick Reference
Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number
More informationLesson 7: Literal Equations, Inequalities, and Absolute Value
, and Absolute Value In this lesson, we first look at literal equations, which are equations that have more than one variable. Many of the formulas we use in everyday life are literal equations. We then
More informationPrep for College Algebra with Trigonometry
Prep for College Algebra with Trigonometry This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (246 topics +
More informationRelationships Between Quantities
Algebra 1 Relationships Between Quantities Relationships Between Quantities Everyone loves math until there are letters (known as variables) in problems!! Do students complain about reading when they come
More informationPart 1 will be selected response. Each selected response item will have 3 or 4 choices.
Items on this review are grouped by Unit and Topic. A calculator is permitted on the Algebra 1 A Semester Exam. The Algebra 1 A Semester Exam will consist of two parts. Part 1 will be selected response.
More informationMath Review for Incoming Geometry Honors Students
Solve each equation. 1. 5x + 8 = 3 + 2(3x 4) 2. 5(2n 3) = 7(3 n) Math Review for Incoming Geometry Honors Students 3. Victoria goes to the mall with $60. She purchases a skirt for $12 and perfume for $35.99.
More informationHow much can they save? Try $1100 in groceries for only $40.
It s Not New, It s Recycled Composition of Functions.4 LEARNING GOALS KEY TERM In this lesson, you will: Perform the composition of two functions graphically and algebraically. Use composition of functions
More informationChapter 3A  Rectangular Coordinate System
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3A! Page61 Chapter 3A  Rectangular Coordinate System A is any set of ordered pairs of real numbers. A relation can be finite: {(3, 1), (3,
More informationSection 4.1 Solving Systems of Linear Inequalities
Section 4.1 Solving Systems of Linear Inequalities Question 1 How do you graph a linear inequality? Question 2 How do you graph a system of linear inequalities? Question 1 How do you graph a linear inequality?
More informationAccelerated Coordinate Algebra/Analytic Geometry A Summer Work (14_15_ACAAGA_SW)
Name: Date: 1. Parallel lines m and n intersect line segment AC and ray CB. Use the information in the diagram to answer the questions. What are the degree measures of and Explain your answers. What is
More informationUNIT #3 LINEAR FUNCTIONS, EQUATIONS, AND THEIR ALGEBRA COMMON CORE ALGEBRA II
Name: Date: Part I Questions UNIT #3 LINEAR FUNCTIONS, EQUATIONS, AND THEIR ALGEBRA COMMON CORE ALGEBRA II. The distance that a person drives at a constant speed varies directly with the amount of time
More informationPre Algebra and Introductory Algebra
Pre Algebra and Introductory Algebra This course covers the topics outlined below and is available for use with integrated, interactive ebooks. You can customize the scope and sequence of this course to
More informationChapter 5 Inequalities
Chapter 5 Inequalities 5.1 Solve inequalities using addition and subtraction 1. Write and graph an inequality. 2. Solve inequalities using addition and subtraction. Review Symbols to KNOW < LESS THAN
More informationName: Class: Date: ID: A. c. the quotient of z and 28 z divided by 28 b. z subtracted from 28 z less than 28
Name: Class: Date: ID: A Review for Final Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Give two ways to write the algebraic expression z 28 in words.
More informationMath 0312 Intermediate Algebra Chapter 1 and 2 Test Review
Math 0312 Intermediate Algebra Chapter 1 and 2 Test Review Name Date MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation for y. 1)
More informationMath Fundamentals for Statistics I (Math 52) Unit 7: Connections (Graphs, Equations and Inequalities)
Math Fundamentals for Statistics I (Math 52) Unit 7: Connections (Graphs, Equations and Inequalities) By Scott Fallstrom and Brent Pickett The How and Whys Guys This work is licensed under a Creative Commons
More informationGrade 8 Mathematics MCA Item Sampler Teacher Guide
Grade 8 Mathematics MCA Item Sampler Teacher Guide Overview of Item Samplers Item samplers are one type of student resource provided to help students and educators prepare for test administration. While
More informationMaintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area
More informationAlgebra II. Note workbook. Chapter 2. Name
Algebra II Note workbook Chapter 2 Name Algebra II: 21 Relations and Functions The table shows the average lifetime and maximum lifetime for some animals. This data can be written as. The ordered pairs
More informationPreAP Algebra 2 Lesson 15 Linear Functions
Lesson 15 Linear Functions Objectives: Students will be able to graph linear functions, recognize different forms of linear functions, and translate linear functions. Students will be able to recognize
More informationMath Scope & Sequence Grades 38
Math Scope & Sequence Grades 38 Texas Essential Knowledge and Skills State Standards Concept/Skill Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Number and Operations in Base Ten Place Value Understand
More informationWillmar Public Schools Curriculum Map
Note: Problem Solving Algebra Prep is an elective credit. It is not a math credit at the high school as its intent is to help students prepare for Algebra by providing students with the opportunity to
More informationMARLBORO CENTRAL SCHOOL DISTRICT CURRICULUM MAP. Unit 1: Integers & Rational Numbers
Timeframe September/ October (5 s) What is an integer? What are some real life situations where integers are used? represent negative and positive numbers on a vertical and horizontal number line? What
More informationAnswer Explanations SAT Practice Test #1
Answer Explanations SAT Practice Test #1 2015 The College Board. College Board, SAT, and the acorn logo are registered trademarks of the College Board. 5KSA09 Section 4: Math Test Calculator QUESTION 1.
More informationAlgebra 1 Math Year at a Glance
Real Operations Equations/Inequalities Relations/Graphing Systems Exponents/Polynomials Quadratics ISTEP+ Radicals Algebra 1 Math Year at a Glance KEY According to the Indiana Department of Education +
More informationPre Algebra. Curriculum (634 topics additional topics)
Pre Algebra This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs.
More informationMath ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying
Math 1050 2 ~ Exam #1 Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying General Tips for Studying: 1. Review this guide, class notes, the
More informationReview for the Algebra EOC
Review for the Algebra EOC The test is Thursday, January 26 th, 2017 The answer key for this review booklet can be found at: www.mrshicklin.pbworks.com 1. A 1,500gallon tank contains 200 gallons of water.
More informationUnit 4: Inequalities. Inequality Symbols. Algebraic Inequality. Compound Inequality. Interval Notation
Section 4.1: Linear Inequalities Section 4.2: Solving Linear Inequalities Section 4.3: Solving Inequalities Applications Section 4.4: Compound Inequalities Section 4.5: Absolute Value Equations and Inequalities
More informationOBJECTIVES UNIT 1. Lesson 1.0
OBJECTIVES UNIT 1 Lesson 1.0 1. Define "set," "element," "finite set," and "infinite set," "empty set," and "null set" and give two examples of each term. 2. Define "subset," "universal set," and "disjoint
More informationFoundations of Algebra. Learning Goal 3.1 Algebraic Expressions. a. Identify the: Variables: Coefficients:
Learning Goal 3.1 Algebraic Expressions What you need to know & be able to do 1. Identifying Parts of Algebraic Expressions 3.1 Test Things to remember Identify Parts of an expression Variable Constant
More informationCore Connections Algebra 2 Checkpoint Materials
Core Connections Algebra 2 Note to Students (and their Teachers) Students master different skills at different speeds. No two students learn eactly the same way at the same time. At some point you will
More informationCommon Core Algebra Regents Review
Common Core Algebra Regents Review Real numbers, properties, and operations: 1) The set of natural numbers is the set of counting numbers. 1,2,3,... { } symbol 2) The set of whole numbers is the set of
More informationNC Common Core Middle School Math Compacted Curriculum 7 th Grade Model 1 (3:2)
NC Common Core Middle School Math Compacted Curriculum 7 th Grade Model 1 (3:2) Analyze proportional relationships and use them to solve realworld and mathematical problems. Proportional Reasoning and
More informationAnswers Investigation 4
Answers Investigation Applications. a. 7 gallons are being pumped out each hour; students may make a table and notice the constant rate of change, which is  7, or they may recognize that  7 is the coefficient
More informationSOLVING LINEAR INEQUALITIES
Topic 15: Solving linear inequalities 65 SOLVING LINEAR INEQUALITIES Lesson 15.1 Inequalities on the number line 15.1 OPENER Consider the inequality x > 7. 1. List five numbers that make the inequality
More informationCOMMON CORE MATHEMATICS CURRICULUM
COMMON CORE MATHEMATICS CURRICULUM Lesson 1 8 4 Lesson 1: Writing Equations Using Symbols Write each of the following statements using symbolic language. 1. When you square five times a number you get
More informationUnit 9: Introduction to Linear Functions
Section 9.1: Linear Functions Section 9.2: Graphing Linear Functions Section 9.3: Interpreting the Slope of a Linear Function Section 9.4: Using Rates of Change to Build Tables and Graphs Section 9.5:
More informationLinear Functions. Unit 3
Linear Functions Unit 3 Standards: 8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and
More informationMATH 081. Diagnostic Review Materials PART 2. Chapters 5 to 7 YOU WILL NOT BE GIVEN A DIAGNOSTIC TEST UNTIL THIS MATERIAL IS RETURNED.
MATH 08 Diagnostic Review Materials PART Chapters 5 to 7 YOU WILL NOT BE GIVEN A DIAGNOSTIC TEST UNTIL THIS MATERIAL IS RETURNED DO NOT WRITE IN THIS MATERIAL Revised Winter 0 PRACTICE TEST: Complete as
More informationIDAHO EXTENDED CONTENT STANDARDS MATHEMATICS
Standard 1: Number and Operation Goal 1.1: Understand and use numbers. K.M.1.1.1A 1.M.1.1.1A Recognize symbolic Indicate recognition of expressions as numbers various # s in environments K.M.1.1.2A Demonstrate
More informationBishop Kelley High School Summer Math Program Course: Algebra 2 A
06 07 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 6 pages of this packet provide eamples as to how to work some of the problems
More informationSection 2.1 Exercises
Section. Linear Functions 47 Section. Exercises. A town's population has been growing linearly. In 00, the population was 45,000, and the population has been growing by 700 people each year. Write an equation
More informationThe graphs of the equations y = 2x and y = 2x + a intersect in Quadrant I for which values of a?
Name: Date: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 The graphs of the equations y = 2x and y = 2x + a intersect in Quadrant I for which values of a? (1) 0 < a < 1 (2) a < 1 (3) a 1 (4) a > 1 20
More informationMathematics 10C. UNIT FIVE Linear Functions. Unit. Student Workbook. y 2.  y 1 x 2. rise. m =  x 1 run. y = mx + b. m = m original. 1 moriginal.
Mathematics 10C Student Workbook Unit m = y 2  y 1 x 2  x 1 run rise Lesson 1: Slope of a Line Approximate Completion Time: 2 Days y = mx + b Lesson 2: SlopeIntercept Form Approximate Completion Time:
More informationEureka Math. Grade, Module. Student _B Contains Sprint and Fluency, Exit Ticket, and Assessment Materials
A Story of Eureka Math Grade, Module Student _B Contains Sprint and Fluency,, and Assessment Materials Published by the nonprofit Great Minds. Copyright 2015 Great Minds. All rights reserved. No part
More information