graphs, Equations, and inequalities 2

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1 graphs, Equations, and inequalities You might think that New York or Los Angeles or Chicago has the busiest airport in the U.S., but actually it s Hartsfield-Jackson Airport in Atlanta, Georgia. In 010, it served over 43 million passengers!.1 the Plane! Modeling Linear Situations What goes Up Must Come Down Analyzing Linear Functions Scouting for Prizes! Modeling Linear Inequalities We re Shipping Out! Solving and Graphing Compound Inequalities Play Ball! Absolute Value Equations and Inequalities Choose Wisely! Understanding Non-Linear Graphs and Inequalities

2 Chapter Overview This chapter reviews solving linear equations and inequalities with an emphasis towards connecting the numeric, graphic, and algebraic methods for solving linear functions. Students explore the advantages and limitations of using tables, functions, and graphs to solve problems. A graphical method for solving linear equations, which involves graphing the left and right side of a linear equation, is introduced. Upon student understanding of solving and graphing equations by hand, the chapter introduces the use of a graphing calculator. Finally, the graphical method for solving problems is extended to include non-linear equations and inequalities. Lesson CCSS Pacing Highlights Models Worked Examples Peer Analysis Talk the Talk Technology.1 Modeling Linear Situations A.REI.1 A.REI.3 A.REI.10 A.CED.1 A.CED. N.Q.1 A.SSE.1.a F.IF. F.IF.6 1 This lesson explores the connections of linear functions represented in a table, as a graph, or in function notation. Questions ask students to compare the contextual meaning of different parts of a linear function to the corresponding mathematical meanings. Students will solve linear functions using function notation and graphs. The concept of graphing both sides of an equation and interpreting the intersection point is introduced as a method to determine solutions to various problem situations. x x x x..3 Analyzing Linear Functions Modeling Linear Inequalities A.REI.3 A.CED.1 A.CED. N.Q.1 A.SSE.1.a A.REI.10 N.Q.3 F.IF. F.IF.6 A.CED.1 A.CED. A.CED.3 A.REI.3 A.REI.10 N.Q.3 1 This lesson provides opportunities for students to explore the use of tables, graphs, and equations to determine solutions. Questions ask students to consider how each representation is used to determine an exact value versus an approximate value. Once students have performed calculations by hand they explore the use of a graphing calculator. The lesson provides graphing calculator instructions to demonstrate how to use technology strategically in determining solutions using various methods. This lesson explores how a negative rate affects solving an inequality. Questions focus students to make connections between graphs on a coordinate plane and solutions represented on a number line. x x x x 71A Chapter Graphs, Equations, and Inequalities

3 Lesson CCSS Pacing Highlights Models Worked Examples Peer Analysis Talk the Talk Technology.4 Solving and Graphing Compound Inequalities A.CED.1 A.CED. A.REI.3 1 This lesson introduces the concept of writing, solving, and graphing compound inequalities. x.5 Absolute Value Equations and Inequalities A.CED.1 A.CED. A.CED.3 A.REI.3 A.REI.10 This lesson provides a review of absolute value as the foundation for solving linear absolute value equations. The lesson then extends the given problem situation to inequalities. Questions focus on the connections between solving linear absolute value inequalities to solving compound inequalities. Students will graph linear absolute value equations on a coordinate plane, use those graphs to answer questions, and then make connections to the algebraic solutions they determined when they solved compound inequalities. x x x x x.6 Understanding Non-Linear Graphs and Inequalities N.Q.1 N.Q. A.CED. A.CED.3 A.REI.10 F.IF. F.LE.1.b F.LE.1.c This lesson provides opportunities for students to solve non-linear problems using technology. Several problem situations are presented and students will choose the correct function to represent each. Questions then guide students to complete a table of values and use a graphing calculator to determine solutions. As a culminating activity, students will describe the advantages and limitations of solving problems numerically, graphically, and algebraically with and without the use of technology. x Chapter Graphs, Equations, and Inequalities 71B

4 Skills Practice Correlation for Chapter Lesson Problem Set Objectives Vocabulary.1 Modeling Linear Situations 1 6 Identify independent and dependent quantities and write functions representing problem situations 7 1 Complete tables of values and calculate unit rates of change Identify input values, output values, and rates of change for given functions 19 4 Solve functions for given input values 5 30 Determine input values for given output values using a graph 1 6 Complete tables to represent problem situations. Analyzing Linear Functions 7 1 Identify input values, output values, y-intercepts, and rates of change for functions Estimate intersection points of functions and dependent values using graphs 19 4 Determine exact values of intersection points using algebra Vocabulary.3 Modeling Linear Inequalities 1 6 Write equations and inequalities using a graph 7 1 Answer questions using a graph and graph solutions on a number line Write and solve inequalities to answer questions from a problem situation 19 4 Represent solutions on a graph by drawing an oval then write corresponding inequality statements Vocabulary 1 6 Write compound inequalities in compact form.4 Solving and Graphing Compound Inequalities 7 1 Write inequalities from number lines Graph inequalities on number lines 19 4 Write compound inequalities representing situations 5 3 Represent solutions of compound inequalities on number lines then write final solutions represented by the number line Solve and graph compound inequalities on number lines 71C Chapter Graphs, Equations, and Inequalities

5 Lesson Problem Set Objectives Vocabulary 1 6 Evaluate absolute values.5 Absolute Value Equations and Inequali ties 7 1 Determine the number of solutions and calculate solutions of equations Solve linear absolute value equations 19 4 Solve linear absolute value equations 5 30 Solve linear absolute value inequalities and graph solutions on number lines Graph functions that represent problem situations and draw an oval to represent solutions.6 Under standing Non-Linear Graphs and Inequalities 1 6 Identify functions that represent problem situations 7 1 Graph functions that represent problem situations and use the graph to answer questions Chapter Graphs, Equations, and Inequalities 71D

6 7 Chapter Graphs, Equations, and Inequalities

7 The Plane! Modeling Linear Situations.1 Learning Goals Key Terms In this lesson, you will: Complete tables and graphs, and write equations to model linear situations. Analyze multiple representations of linear relationships. Identify units of measure associated with linear relationships. Determine solutions both graphically and algebraically. Determine solutions to linear functions using intersection points. first differences solution intersection point Essential Ideas Multiple representations such as tables, graphs, and equations are used to model linear situations. Solutions to linear functions are determined both graphically and algebraically. Intersection points are used to determine solutions to linear functions. Solve equations and inequalities in one variable 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Represent and solve equations and inequalities graphically Mathematics Common Core Standards A-REI Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-CED Creating Equations Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 73A

8 N-Q Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. A-SSE Seeing Structure in Expressions Interpret the structure of expressions. 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. F-IF Interpreting Functions Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Interpret functions that arise in applications in terms of the context 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Overview Students are given a scenario and asked to represent the situation using tables, graphs, and expressions. They will determine the unit rate of change using two different methods, and then use the graph to identify the slope and the x- and y-intercepts. Students will also compare the contextual meaning of the various parts of the expression to the mathematical meaning. The expression is expressed using function notation and students will solve the equation for several inputs relating each output to the problem situation. Next, students are given output values, and solve for input values both algebraically and graphically. The concept of intersecting lines and the point of intersection is used to identify solutions. The graphs of the equations written in function notation of the intersection lines are related to the scenario and possible outputs, and the students are asked to identify the inputs as the x-coordinates of the points of intersection. Students will discuss how to determine if a function is linear given a table, a graph, or an equation. They will then discuss advantages and disadvantages of solving equations algebraically and graphically. 73B Chapter Graphs, Equations, and Inequalities

9 Warm Up A tree grows at a rate of 3.5 feet per year. 1. Identify the independent and dependent quantities and their unit of measure in this problem situation. The independent quantity is time measure in years. The dependent quantity is the height of the tree measured in feet.. Suppose t represents the time in terms of years and h(t) represents the height of the tree in terms of feet over a period of time. Complete a table of values to describe this situation. t (years) h(t) (feet) Write an equation in function notation to represent the problem situation. h(t) 5 3.5t 4. Sketch the graph of the problem situation and label the axes. h(t) Tree Growth Height (inches) Time (years) t.1 Modeling Linear Situations 73C

10 73D Chapter Graphs, Equations, and Inequalities

11 the Plane! Modeling linear Situations.1 learning goals In this lesson, you will: Complete tables and graphs, and write equations to model linear situations. Analyze multiple representations of linear relationships. Identify units of measure associated with linear relationships. Determine solutions both graphically and algebraically. Determine solutions to linear functions using intersection points. KEy terms first differences solution intersection point adies and gentlemen, at this time we ask that all cell phones and pagers be L turned off for the duration of the flight. All other electronic devices must be turned off until the aircraft reaches 10,000 feet. We will notify you when it is safe to use such devices. Flight attendants routinely make announcements like this on airplanes shortly before takeoff and landing. But what s so special about 10,000 feet? When a commercial airplane is at or below 10,000 feet, it is commonly known as a critical phase of flight. This is because research has shown that most accidents happen during this phase of the flight either takeoff or landing. During critical phases of flight, the pilots and crew members are not allowed to perform any duties that are not absolutely essential to operating the airplane safely. And it is still not known how much interference cell phones cause to a plane s instruments. So, to play it safe, crews will ask you to turn them off Modeling Linear Situations 73

12 Problem 1 Given a scenario, students will identify the independent and dependent quantities and their units of measure, and the appropriate function family. Then they sketch a simple graph of the situation labeling the axes with the independent and dependent quantities and their units of measure. Next, they will create a table of values, calculate several outputs, and write an expression to represent the dependent quantity. Students will explore the first differences as a way to identify if the situation represents a linear function. Students conclude the rates of change of a linear function must be constant. PROBLEM 1 Analyzing tables A 747 airliner has an initial climb rate of 1800 feet per minute until it reaches a height of 10,000 feet. 1. Identify the independent and dependent quantities in this problem situation. Explain your reasoning. The height of the airplane depends on the time, so height is the dependent quantity and time is the independent quantity.. Describe the units of measure for: a. the independent quantity (the input values). The independent quantity of time is measured in minutes. b. the dependent quantity (the output values). The dependent quantity of height is measured in feet. 3. Which function family do you think best represents this situation? Explain your reasoning. Answers will vary. The situation shows a linear function because the rate the plane ascends is constant. So, this situation belongs to the linear function family. Grouping Ask a student to read the scenario. Discuss and complete Questions 1 and as a class. Have students complete Questions 3 and 4 independently. Then share the responses as a class. 4. Draw and label two axes with the independent and dependent quantities and their units of measure. Then sketch a simple graph of the function represented by the situation. Height (feet) y Time (minutes) x When you sketch a graph, include the axes labels and the general graphical behavior. Be sure to consider any intercepts. Guiding Questions for Discuss Phase, Questions 1 and Is the airliner ascending or descending in the problem situation? What is the maximum height of the airliner? Is the airliner ascending at a constant rate? What is the constant rate at which the airliner ascends? Guiding Questions for Share Phase, Questions 3 and 4 What are the characteristics of the linear family of functions? Should the independent quantity or the dependent quantity be placed on the x-axis? Should the independent quantity or the dependent quantity be placed on the y-axis? Does your graph pass through the origin? How would you describe the graphical behavior of this situation? 74 Chapter Graphs, Equations, and Inequalities

13 Grouping Have students complete Questions 5 and 6 with a partner. Then share the responses as a class. Ask a student to read the information before Question 7. Discuss as a class. Guiding Questions for Share Phase, Questions 5 and 6 How did you determine the height of the airliner when the time was equal to 1 minute? How did you determine the height of the airliner when the time was equal to minutes? How did you determine the height of the airliner when the time was equal to.5 minutes? How did you determine the height of the airliner when the time was equal to 3 minutes? How did you determine the height of the airliner when the time was equal to t minutes? How many minutes would it take for the airliner to reach the maximum height? Although it is a convention to place the independent quantity on the left side of the table, it really dœsn t matter. 5. Write the independent and dependent quantities and their units of measure in the table. Then, calculate the dependent quantity values for each of the independent quantity values given. Independent Quantity Dependent Quantity Quantity Time Height Units minutes feet Expression t 1800t 6. Write an expression in the last row of the table to represent the dependent quantity. Explain how you determined the expression. Each input of time is multiplied by 1800 to produce a height in feet, so the expression for the dependent quantity is 1800t. Why do you think t was chosen as the variable?.1 Modeling Linear Situations 7.1 Modeling Linear Situations 75

14 Grouping Have students complete Questions 7 and 8 with a partner. Then share the responses as a class. Have students complete Questions 9 through 13 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Questions 7 and 8 Why is it important to calculate the first differences? If the input values are not consecutive whole numbers, is it still possible to calculate the first differences? Why or why not? What is the unit rate of change for this situation? Is this situation linear in nature? Guiding Questions for Share Phase, Question 9 When do you need to calculate the unit rate of change differently? How is this second method of calculating the unit rate of change different than the previous method? When should you use it? Is the difference between the output values placed in the numerator or the denominator of the rate of change? Let s examine the table to determine the unit rate of change for this situation. One way to determine the unit rate of change is to calculate first differences. Recall that first differences are determined by calculating the difference between successive points. 7. Determine the first differences in the section of the table shown Time (minutes) Height (feet) First Differences What do you notice about the first differences in the table? Explain what this means. The first differences are all the same. They are all This means that the unit rate of change is 1800 feet per minute. Another way to determine the unit rate of change is to calculate the rate of change between any two ordered pairs and then write each rate with a denominator of Calculate the rate of change between the points represented by the given ordered pairs in the section of the table shown. Show your work. These numbers are not consecutive. I wonder if that is why I have to use another method. a. (.5, 4500) and (3, 5400) b. (3, 5400) and (5, 9000) c. (.5, 4500) and (5, 9000) Time (minutes) Height (feet) Remember, if you have two ordered pairs, the rate of change is the difference between the output values over the difference between the input values. Is the difference between the input values placed in the numerator or the denominator of the rate of change? Why must the unit rate of change have 1 in the denominator? Is the rate of change in this situation constant? 76 Chapter Graphs, Equations, and Inequalities

15 Guiding Questions for Share Phase, Questions 10 through 13 If the rate of change is constant, how would you describe the graphical behaviors of the situation? How many miles per hour is 1800 feet per minute? Is height the same thing as horizontal distance? What is the difference between height and horizontal distance? Is the 1800 feet per minute a description of height or horizontal distance? Is a car traveling down a road best described by height or horizontal distance? Is the ground speed of a plane faster or slower than the ground speed of a car? 10. What do you notice about the rates of change? The rate of change is 1800 feet per minute given any two points in the table. 11. Use your answers from Question 7 through Question 10 to describe the difference between a rate of change and a unit rate of change. The rate of change for the plane s climb can be written as 1800 or 900, and so on The unit rate of change must have a denominator of 1. So, the unit rate of change is How do the first differences and the rates of change between ordered pairs demonstrate that the situation represents a linear function? Explain your reasoning. Both show that the rate of change in the function is constant. When the rate of change between all points is constant, the ordered pairs will form a straight line when plotted.? 13. Alita says that in order for a car to keep up with the plane on the ground, it would have to travel at only 0.5 miles per hour. Is Alita correct? Why or why not? Alita is not correct. It is true that 1800 feet per minute is about 0.5 miles per hour, but this rate compares height to time, not horizontal distance to time. The plane is ascending at about 0.5 miles per hour, but its horizontal speed, or ground speed, is probably much faster..1 Modeling Linear Situations 7.1 Modeling Linear Situations 77

16 Problem Students are given three parts of the expression used in the scenario of the last problem. They will write the appropriate unit of measure associated with each expression and describe what each part means in terms of the problem situation and in terms of an input value, output value, or rate of change. Next, they use a function in the form h(t) to describe the plane s height over time, t. After graphing the function, they will identify the slope, and the intercepts. The function is then used to determine the height of the plane at several given times, and students interpret each solution in terms of the problem situation. PROBLEM Analyzing Equations and graphs 1. Complete the table shown for the problem situation described in Problem 1, Analyzing Tables. First, determine the unit of measure for each expression. Then, describe the contextual meaning of each part of the function. Finally, choose a term from the word box to describe the mathematical meaning of each part of the function. What It Means Expression Unit Contextual Meaning Mathematical Meaning t minutes 1800 feet minute output value input value rate of change the time, in minutes, that the plane has been in the air the number of feet that the plane climbs each minute input value rate of change 1800t feet the height, in feet, of the plane output value Grouping Have students complete Questions 1 through 4 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Questions 1 through 3 What is the expression t associated with in the scenario in Problem 1? What is the expression 1800 associated with in the scenario in Problem 1? What is the expression 1800t associated with in the scenario in Problem 1? How do you know which expression is the input value and which expression is the output value?. Write a function, h(t), to describe the plane s height over time, t. h(t) t 3. Which function family does h(t ) belong to? Is this what you predicted back in Problem 1, Question 3? It is a linear function because it is in the form f(x) 5 mx 1 b. Yes. I predicted that the function would be linear because the plane s ascent is constant over time. Why do you think h(t) is used to name this function? What does h(t) represent with respect to the problem situation? Is the function h(t) t written in the form of f(x) 5 ax 1 b? What is a? What is b? 78 Chapter Graphs, Equations, and Inequalities

17 Guiding Questions for Share Phase, Question 4 What is the independent quantity? Which axis is associated with this quantity? What is the dependent quantity? Which axis is associated with this quantity? How would you describe the domain of this graph? Of this function? How would you describe the range of this graph? Of this function? Are there points on this function that are not visible on this graph? Where? Is this considered a complete graph? Why or why not? What does the slope represent in terms of this problem situation? 4. Use your table and function to create a graph to represent the change in the plane s height as a function of time. Be sure to label your axes with the correct units of measure and write the function. Height (feet) y h(t) t Time (minutes) a. What is the slope of this graph? Explain how you know. The equation is h(t) t. The slope of the graph is In slope-intercept form y 5 mx 1 b, the slope is given by the coefficient of the x-variable, or m. The slope is the same as the constant unit rate of change, which is b. What is the x-intercept of this graph? What is the y-intercept? Explain how you determined each intercept. The x-intercept is 0, and the y-intercept is 0. The x-intercept is the x-coordinate of the point where the line crosses the x-axis. The graph crosses the x-axis at x 5 0. The y-intercept is the y-coordinate of the point where the line crosses the y-axis. The graph crosses the y-axis at y 5 0. x c. What do the x- and y-intercepts mean in terms of this problem situation? The x-intercept represents 0 minutes, the time when the plane takes off. The y-intercept represents 0 feet, the height of the plane just as it takes off..1 Modeling Linear Situations 7.1 Modeling Linear Situations 79

18 Grouping Ask a student to read the Worked Example and complete Question 5 as a class. Have students complete Question 6 with a partner. Then share the responses as a class. Let s consider how to determine the height of the plane, given a time in minutes, using function notation. To determine the height of the plane at minutes using your function, substitute for t every time you see it. Then, simplify the function. h(t) t Substitute for t. h() () h() Guiding Questions for Discuss Phase, Question 5 Does h(t) represent the height of the plane? Does h() represent the height of the plane? Does 1800() represent the height of the plane? Does 3600 represent the height of the plane? Guiding Questions for Discuss Phase, Question 6 What would h(0) represent with respect to this problem situation? What would h(positive number) represent with respect to the problem situation? What would h(negative number) represent with respect to the problem situation? How did you calculate h(3)? How did you calculate h(3.75)? How did you calculate h(5.1)? How did you calculate h(4)? 5. List the different ways the height of the plane is represented in the example. The different ways the height of the plane is represented are h(t), h(), 1800(), and Use your function to determine the height of the plane at each given time in minutes. Write a complete sentence to interpret your solution in terms of the problem situation. a. h(3) 5 Two minutes after takeoff, the plane is at 3600 feet ft h(3) (3) The plane is at 5400 feet 3 minutes after takeoff ft c. h(5.1) 5 h(5.1) (5.1) The plane is at 9180 feet 5.1 minutes after takeoff ft b. h(3.75) 5 h(3.75) (3.75) The plane is at 6750 feet 3.75 minutes after takeoff. 700 ft d. h( 4) 5 h( 4) (4) If I evaluate the function at 4, I get 700 feet. In terms of the problem situation, 4 would mean 4 minutes ago. In this context, negative height does not make sense. The plane is on the runway at 0 feet. 80 Chapter Graphs, Equations, and Inequalities

19 Problem 3 Using the scenario from Problem 1 and the equation describing the scenario written in function notation from Problem, the students will determine the input value given the output value. The term solution of a linear equation is described as any value that makes the open sentence true and, graphically speaking, the solution is any point on the line. If a graph contains intersecting lines, the solution is the ordered pair that satisfies both equations and the point at which both lines intersect. An example of this is provided and students interpret the point of intersection with respect to this problem situation. Finally, students will use graphs of intersecting lines to determine the input values by identifying the x-coordinates of the point of intersection. Grouping Ask a student to read the introduction. Discuss the worked example and complete Question 1 as a class. Have students complete Question with a partner. Then share the responses as a class. Guiding Questions for Discuss Phase, Question 1 What operation was used to determine the time given the height? What does 700 represent with respect to the problem situation? PROBLEM 3 Connecting Approaches Now let s consider how to determine the number of minutes the plane has been flying (the input value) given a height in feet (the output value) using function notation. To determine the number of minutes it takes the plane to reach 700 feet using your function, substitute 700 for h(t) and solve. Substitute 700 for h(t). 1. Why can you substitute 700 for h(t)? Both 700 and h(t) represent the height of the plane. Therefore, they are equivalent.. Use your function to determine the time it will take the plane to reach each given height in feet. Write a complete sentence to interpret your solution in terms of the problem situation. a feet t t t The plane will be at 5400 feet 3 minutes after takeoff. c feet t t t The plane will be at 3150 feet 1.75 minutes after takeoff. h(t) t t t t After takeoff, it takes the plane 4 minutes to reach a height of 700 feet. b feet t t t The plane will be at 9000 feet 5 minutes after takeoff. d feet t t t The plane will be at 4500 feet.5 minutes after takeoff. What does h(t) represent with respect to the problem situation?.1 Modeling Linear Situations 81 Guiding Questions for Share Phase, Question Is 5400 feet considered the input or the output? What equation did you use to solve for the time? How many steps did it take to solve the equation? What operation was used to solve the equation? Is.5 minutes equal to minutes and 5 seconds or is it equal to minutes and 30 seconds?.1 Modeling Linear Situations 81

20 Grouping Ask a student to read the definition. Discuss the worked example and complete Question 3 as a class. Have students complete Questions 4 and 5 with a partner. Then share the responses as a class. You can also use the graph to determine the number of minutes the plane has been flying (input value) given a height in feet (output value). Remember, the solution of a linear equation is any value that makes the open sentence true. If you are given a graph of a function, a solution is any point on that graph. The graph of any function, f, is the graph of the equation y 5 f(x). If you have intersecting graphs, a solution is the ordered pair that satisfies both functions at the same time, or the intersection point of the graphs. To determine how many minutes it takes for the plane to reach 700 feet using your graph, you need to determine the intersection of the two graphs represented by the equation t. Guiding Questions for Discuss Phase, Question 3 What does the point (1, 1800) on the graph represent with respect to the problem situation? What does the point (, 3600) on the graph represent with respect to the problem situation? What does the point (3, 5400) on the graph represent with respect to the problem situation? What does the point (4, 700) on the graph represent with respect to the problem situation? What does the point (5, 9000) on the graph represent with respect to the problem situation? Guiding Questions for Share Phase, Question 4 What does h(t) represent with respect to the problem situation? First, graph each side of the equation and then determine the intersection point of the two graphs. Height (feet) y h(t) t y (4, 700) (t, h(t)) Time (minutes) After takeoff, it takes the plane 4 minutes to reach a height of 700 feet. 3. What does (t, h(t)) represent? The ordered pair (t, h(t)) satisfies both equations at the same time. It is the intersection point of the two equations. 4. Explain the connection between the form of the function h(t) t and the equation y x in terms of the independent and dependent quantities. The dependent quantities are represented by h(t) and y in each form. The independent quantities are represented by t and x in each form. Are the solutions you determined algebraically in the last problem in support of the solutions you determined graphically in this problem? x h(t) t t y y x Solution: (4, 700) 8 Chapter Graphs, Equations, and Inequalities

21 Guiding Questions for Share Phase, Question 5 Which solutions were difficult to read on the graph? Why? When is it best to use a graph to solve for a solution? When is it best to use algebra to solve for a solution? 5. Use the graph to determine how many minutes it will take the plane to reach each height. a. h(t) b. h(t) c. h(t) d. h(t) Height (feet) y h(t) t (5, 9000) (3, 5400) (.5, 4500) (1.75, 3150) Time (minutes) x y y y y Label all your horizontal lines and the intersection points. Were you able to get exact answers using the graph? 6. Compare and contrast your solutions using the graphing method to the solutions in Question, parts (a) through (d) where you used an algebraic method. What do you notice? The solutions are the same. Some solutions are hard to read on the graph, but solving the equation for the input value gives me an exact answer..1 Modeling Linear Situations 8.1 Modeling Linear Situations 83

22 Talk the Talk Students will describe how to analyze a linear function represented in a table, a graph, or an equation. They then identify possible advantages and disadvantages of each representation with respect to a graphic or algebraic method of solution. Finally, they will solve four equations using the Properties of Equality. talk the talk You just worked with different representations of a linear function. 1. Describe how a linear function is represented: a. in a table. When the input values in a table are in successive order and the first differences of the output values are constant, the table represents a linear function. b. in a graph. A linear function is represented in a graph by a straight line. Grouping Ask a student to read the problem. Discuss and complete Questions 1 through 3 as a class. Guiding Questions for Discuss Phase, Questions 1 through 3 What does a table of values describing a linear function look like? What does a graph describing a linear function look like? What does an equation describing a linear function look like? Which representation is easiest to work with? Why? Which representation is most difficult to work with? Why? If given the choice, which representation do you prefer to use? Why? Do you think all three representations can be used with functions that are not linear in nature? Explain. c. in an equation. A linear function is represented by a function in the form f(x) 5 ax 1 b.. Name some advantages and disadvantages of the graphing method and the algebraic method when determining solutions for linear functions. Answers will vary. Graphs provide visual representations of functions, and they can provide a wide range of values, depending on the intervals. A disadvantage is that I have to estimate values if points do not fall exactly on grid line intersections. The algebraic method provides an exact solution for every input, but I may be unable to solve more difficult equations correctly. 3. Do you think the graphing method for determining solutions will work for any function? Answers will vary. No. The graphing method would probably be too difficult to use for complicated functions. 84 Chapter Graphs, Equations, and Inequalities

23 Grouping Have the students complete Question 4 with a partner. Share the responses as a class. You solved several linear equations in this lesson. Remember, the Addition, Subtraction, Multiplication, and Division Properties of Equality allow you to balance and solve equations. The Distributive Property allows you to rewrite expressions to remove parentheses, and the Commutative and Associative Properties allow you to rearrange and regroup expressions. Guiding Questions for Shares Phase, Question 4 How many steps did it take to solve the problem? What was the first step? Is there more than one way to solve the problem? Explain. How do you check your answer? 4. Solve each equation and justify your reasoning. a. b. 7x (x 1 7) x x x x x 5 4x x x x 5 9 c. 14x x x x x 5 9x x 9x 5 9x 9x x x x d. x (x 1 ) 5 6? 5x x x 5 x 5 5 Don t forget to check your answers. Be prepared to share your solutions and methods..1 Modeling Linear Situations 8.1 Modeling Linear Situations 85

24 Check for Students Understanding Roman shovels snow for winter work and charges $0 per sidewalk. 1. Identify the independent and dependent quantities in this problem situation. The independent quantity is number of sidewalks shoveled. The dependent quantity is the money Rowan makes.. Suppose s represents the number of sidewalks Roman shoveled and P(s) represents the profit Roman makes shoveling sidewalks. Complete a table of values to describe this situation. s (sidewalks) P(s) (dollars) Does the table of values describe a function? Why or why not? The table of values describes a function because there is exactly one y-value for each x-value. 4. Write an equation in function notation to represent this problem situation. p(s) 5 0s 86 Chapter Graphs, Equations, and Inequalities

25 5. Sketch the graph of this problem situation and label the axes. p(s) Winter Work Profit (dollars) Sidewalks (number of sidewalks) s 6. Is your graph continuous or discrete? Explain your reasoning. The graph is discrete because Roman cannot shovel fractions of sidewalks and the profit made is a multiple of $0. 7. What family of functions is best associated with this problem situation, table, equation and graph? The graph, table, and equation describe a linear relationship. The unit rate of change is constant..1 Modeling Linear Situations 86A

26 86B Chapter Graphs, Equations, and Inequalities

27 What Goes Up Must Come Down Analyzing Linear Functions. Learning Goals In this lesson, you will: Complete tables and graphs, and write equations to model linear situations. Analyze multiple representations of linear relationships. Identify units of measure associated with linear relationships. Determine solutions to linear functions using intersection points and properties of equality. Determine solutions using tables, graphs, and functions. Compare and contrast different problem-solving methods. Estimate solutions to linear functions. Use a graphing calculator to analyze functions and their graphs. Essential Ideas Multiple representations such as tables, graphs, and equations are used to model linear situations. Solutions to linear functions are determined both graphically and algebraically. Intersecting points are used to determine solutions to linear functions. The graphing calculator functions such as table, table set, value, and intersection are used to solve for unknowns. A-CED Creating Equations Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Mathematics Common Core Standards A-REI Reasoning with Equations and Inequalities Solve equations and inequalities in one variable 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. N-Q Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 87A

28 A-SSE Seeing Structure in Expressions Interpret the structure of expressions. 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. A-REI Reasoning with Equations and Inequalities Represent and solve equations and inequalities graphically 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). N-Q Quantities Reason quantitatively and use units to solve problems. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. F-IF Interpreting Functions Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Interpret functions that arise in applications in terms of the context 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Overview Students are given a linear scenario with a negative slope. They will represent the situation using a table, a graph, and an equation. They also identify the independent and dependent quantities, the slope, and the intercepts with respect to the problem situation. Students will determine solutions to this problem situation using each representation and then compare and contrast each solution. The solution is determined both algebraically and graphically. Graphing calculator functions such as the table, table set, value, and intersect functions are introduced, and used to solve for unknowns. 87B Chapter Graphs, Equations, and Inequalities

29 Warm Up Use the graph to answer each question. y How Much Juice Is Left? Juice in Container (fluid ounces) (0, 64) (1, 56) (, 48) (3, 40) (4, 3) (5, 4) (6, 16) (7, 8) (8, 0) (9, 8) x Cups of Juice Consumed (cups) 1. Describe the scenario to fit the graph. The graph describes the number of fluid ounces of juice remaining in a container as a number of 8 fluid ounces servings are consumed.. What is the independent variable? The independent variable is the number of 8 fluid ounce servings of juice consumed. 3. What is the dependent variable? The dependent variable is the amount of juice (fluid ounces) remaining in the container. 4. How does the amount of juice remaining in the container change as each cup of juice is removed? Explain your reasoning. As the number of servings removed increases by one, the amount of juice remaining in the container decreases by 8 fluid ounces because one cup of juice removes 8 fluid ounces. 5. Is the relation a function? Explain your reasoning. The relation is a function because for every input there is exactly one output. 6. Is the graph continuous or discrete? The graph is discrete because each serving is exactly 8 ounces.. Analyzing Linear Functions 87C

30 7. Write a function f(x) that represents the amount of juice in fluid ounces remaining in the container and x represents the number of 8 ounce cups consumed. f(x) x 8. Use your function to determine the amount of juice remaining in the container if 9 cups of juice are consumed. Does your answer make sense? f(x) (9) 5 8 If 9 cups of juice are consumed, there would be 8 fluid ounces of juice remaining in the container which makes no sense. The container has a maximum amount of 64 fluid ounces and to remove 9 cups the container would need to hold 7 fluid ounces. 9. Use your equation to determine the number of cups of juice consumed if the remaining amount of juice is 40 fluid ounces (x) x 5 3 If there were 40 fluid ounces of juice remaining in the container, then 3 cups of juice were consumed. 87D Chapter Graphs, Equations, and Inequalities

31 What goes Up Must Come Down Analyzing linear Functions learning goals. In this lesson, you will: Complete tables and graphs, and write equations to model linear situations. Analyze multiple representations of linear relationships. Identify units of measure associated with linear relationships. Determine solutions to linear functions using intersection points and properties of equality. Determine solutions using tables, graphs, and functions. Compare and contrast different problem-solving methods. Estimate solutions to linear functions. Use a graphing calculator to analyze functions and their graphs. he dollar is just one example of currency used around the world. For example, T Swedes use the krona, Cubans use the peso, and the Japanese use the yen. This means that if you travel to another country you will most likely need to exchange your U.S. dollars for a different currency. The exchange rate represents the value of one country s currency in terms of another and it is changing all the time. In some countries, the U.S. dollar is worth more. In other countries, the dollar is not worth as much. Why would knowing the currency of another country and the exchange rate be important when planning trips?. Analyzing Linear Functions 87

32 Problem 1 A scenario is given with a negative slope that is similar to the scenario in the previous lesson that had a positive slope. Students will identify the independent and dependent quantities including their units of measure, and the rate of change. They then complete a table of values and calculate several inputs and outputs. The values in the table are used to write an expression to represent the problem situation, and the expression is used to write a function. Several expressions are then given and students identify the input value, the output value, the x-intercept, the y-intercept and the rate of change in terms of their units of measure, contextual meaning, and mathematical meaning. Students will sketch a simple graph of the situation labeling the axes with the independent and dependent quantities and their units of measure. Students calculate several outputs both algebraically and graphically and compare the results. Exact answers are distinguished from approximate answers with respect to the multiple representations. Grouping Ask a student to read the scenario. Discuss and and complete Question 1 as a class. Have students complete Questions through 5 with a partner. Then share the responses as a class. PROBLEM 1 As We Make Our Final Descent At 36,000 feet, the crew aboard the 747 airplane begins making preparations to land. The plane descends at a rate of 1500 feet per minute until it lands. 1. Compare this problem situation to the problem situation in Lesson.1, The Plane! How are the situations the same? How are they different? The independent quantity and dependent quantity are the same with time being the independent quantity and height being the dependent quantity in both problem situations. The rate of change is different for each situation. In this situation, the plane is now descending as opposed to taking off.. Complete the table to represent this problem situation. Independent Quantity Dependent Quantity Quantity Time Height Units minutes feet 0 36,000 33, , , , Expression t 1500t 1 36, Write a function, g(t), to represent this problem situation. g(t) t 1 36,000 Guiding Questions for Discuss Phase, Question 1 Is the airliner ascending or descending in the problem situation? What is the height of the airliner before the descent? Is the airliner descending at a constant rate? What is the constant rate at which the airliner descends? Think about the pattern you used to calculate each dependent quantity value. 88 Chapter Graphs, Equations, and Inequalities

33 Guiding Questions for Share Phase, Questions through 5 How did you determine the height of the airliner when the time was equal to t minutes? How many minutes would it take for the airliner to reach the ground? How did writing the expression help when writing the function? What is the difference between writing an expression and writing a function? How do you know which expression is the input value and which expression is the output value? What does g(t) represent with respect to this problem situation? Is the function g(t) t 1 36,000 written in the form of f(x) 5 ax 1 b? What is a? What is b? What is the independent quantity? Which axis is associated with this quantity? What is the dependent quantity? Which axis is associated with this quantity? How would you describe the domain of this graph? Of this function? How would you describe the range of this graph? Of this function? Are there points on this function that are not visible on this graph? Where? 4. Complete the table shown. First, determine the unit of measure for each expression. Then, describe the contextual meaning of each part of the function. Finally, choose a term from the word box to describe the mathematical meaning of each part of the function. input value output value rate of change Expression t y-intercept Units minutes 1500 feet minute 1500t x-intercept Contextual Meaning Description the time, in minutes, that the plane descends the number of feet that the plane descends each minute Is this considered a complete graph? Why or why not? What does the slope represent in terms of the problem situation? Does your graph pass through the origin? Why or why not? How would you describe the graphical behavior of this situation? feet 36,000 feet 1500t 1 36,000 feet 5. Graph g(t) on the coordinate plane shown. Height (feet) 36,000 3,000 8,000 4,000 0,000 16,000 1, y g(t) t 1 36,000 the number of feet the plane has descended the height of the plane, in feet, before it starts its descent the height, in feet, of the plane Time (minutes) x y 5 4,000 y 5 14,000 Mathematical Meaning input value rate of change y-intercept output value. Analyzing Linear Functions 8. Analyzing Linear Functions 89

34 Grouping Ask a student to read the information aloud. Discuss as a class. Have students complete Questions 6 through 8 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Questions 6 Using the table of values, between what two times is the airliner 14,000 feet above the ground? Do the two lines on the graph intersect at a point closer to 1 minutes or closer to 0 minutes? Do the two lines on the graph intersect at a point closer to 1 minutes or closer to 16 minutes? Does the intersection of the two lines on the graph look like it is halfway between 1 minutes and 16 minutes? What equation was used to solve for t? How many steps were used to solve the equation? What was the first step in the solution of the equation? Which method gave you an exact answer, using the table, using the graph, or solving the equation? You have just represented the As We Make Our Final Descent scenario in different ways: numerically, by completing a table, algebraically, by writing a function, and graphically, by plotting points. Let s consider how to use each of these representations to answer questions about the problem situation. 6. Determine how long will it take the plane to descend to 14,000 feet. a. Use the table to determine how long it will take the plane to descend to 14,000 feet. The plane will descend to 14,000 feet between 1 and 0 minutes. The plane will descend to 14,000 feet in approximately 14 minutes. b. Graph and label y 5 14,000 on the coordinate plane. Then determine the intersection point. Explain what the intersection point means in terms of this problem situation. See graph. The graph of y 5 14,000 crosses the graph of g(t) t 1 36,000 about halfway between t 5 1 and t I estimate that it will take the plane about 14 minutes to reach 14,000 feet because that is halfway between 1 and 16. c. Substitute 14,000 for g(t) and solve the equation for t. Interpret your solution in terms of this problem situation. g(t) t 1 36,000 14, t 1 36,000 14,000 36, t 1 36,000 36,000, t t (14.667, 14,000) At about minutes, the plane is at 14,000 feet. d. Compare and contrast your solutions using the table, graph, and the function. What do you notice? Explain your reasoning. Using the table, I can only estimate the time. When plotting the points on my graph, I was able to approximate the intersection of the graphs of both functions. When I used the function, I was able to determine an exact time when the plane would descend to 14,000 feet. 90 Chapter Graphs, Equations, and Inequalities

35 Guiding Questions for Share Phase, Question 7 Using the table of values, between what two times is the airliner 4,000 feet above the ground? Do the two lines on the graph intersect at a point closer to 6 minutes or closer to 1 minutes? Does the intersection of the two lines on the graph look like it is halfway between 4 minutes and 1 minutes? What equation was used to solve for t? How many steps were used to solve the equation? What was the first step in the solution of the equation? Which method gave you an exact answer, using the table, using the graph, or solving the equation? 7. Determine how long it will take the plane to descend to 4,000 feet. a. Use the table to determine how long it will take the plane to descend to 4,000 feet. The plane will descend to 4,000 feet between 6 and 1 minutes. b. Graph and label y = 4,000 on the coordinate plane. Then determine the intersection point. Explain what the intersection point means in terms of this situation. See graph. The graph of y 5 4,000 crosses the original graph at t 5 8. The intersection point means that at 8 minutes, the plane is at 4,000 feet. c. Substitute 4,000 for g(t) and solve the equation for t. Interpret your solution in terms of this situation. g(t) t 1 36,000 4, t 1 36,000 4,000 36, t 1 36,000 36,000 1, t t At 8 minutes, the plane is at 4,000 feet. d. Compare and contrast your solutions using the table, graph, and the function. What do you notice? Explain your reasoning. Using the table, I can only estimate the time. When plotting the points on my graph, I was able to determine the time because it crossed exactly at a marked point on the coordinate plane. When I used the function, I was able to confirm an exact time when the plane would descend to 4,000 feet.. Analyzing Linear Functions 9. Analyzing Linear Functions 91

36 Guiding Questions for Share Phase, Question 8 What are the advantages and disadvantages of using a table to calculate the time or height? What are the advantages and disadvantages of viewing a graph to calculate the time or height? What are the advantages and disadvantages of solving an equation to calculate the time or height? Grouping Have students complete Question 9 independently. Then share the responses as a class. 8. For how many heights can you calculate the exact time using the: a. table? Answers will vary. There are 6 rows of values in the table, so I can use the table to calculate the exact time for 6 different heights. b. graph? Answers will vary. There are 4 instances where the graph of the function crosses an intersection of grid lines on the coordinate plane, so I can use the graph to calculate 4 different exact times. c. function? Answers will vary. I can solve the function for time given any height, so I can calculate an infinite number of exact times given the height. 9. Use the word bank to complete each sentence. Guiding Questions for Share Phase, Question 9 What is an exact answer? What is an approximate answer? What is the difference between an exact answer and an approximate answer? What is an example of when it important to have an exact answer? What is an example of when it acceptable to have an approximate answer? always sometimes never If I am given a dependent value and need to calculate an independent value of a linear function, a. I can always use a table to determine an approximate value. b. I can sometimes use a table to calculate an exact value. c. I can always use a graph to determine an approximate value. d. I can sometimes use a graph to calculate an exact value. e. I can always use a function to determine an approximate value. f. I can always use a function to calculate an exact value. 9 Chapter Graphs, Equations, and Inequalities

37 Problem A scenario is given and students will define variables and use them to write a function to represent the situation. Next, they identify the slope and intercepts and relate them to the problem situation. Students also solve a Who s Correct problem within the context of the scenario using algebraic solutions. The focus of the problem is possible rounding errors that result from computations. PROBLEM Making the Exchange The plane has landed in the United Kingdom and the Foreign Language Club is ready for their adventure. Each student on the trip boarded the plane with 300. They each brought additional U.S. dollars with them to exchange as needed. The exchange rate from U.S. dollars to British pounds is pound to every dollar. 1. Write a function to represent the total amount of money in British pounds each student will have after exchanging additional U.S. currency. Define your variables. Let d represent the number of additional U.S. dollars to exchange, and f(d ) represent the total amount of British pounds each student has. f(d ) d The symbol means pounds, just like $ means dollars. Grouping Have students complete Questions 1 through 5 with a partner. Then share the responses as a class.. Identify the slope and interpret its meaning in terms of this problem situation. The slope is The slope represents the amount in British pounds for 1 U.S. dollar. Guiding Questions for Share Phase, Questions 1 through 3 What variable was used to represent the independent quantity? What variable was used to represent the dependent quantity? Is your function written using function notation? How can you tell? Is the function written in y 5 ax 1 b form? What is the a? What is the b? What function family is associated with this function? 3. Identify the y-intercept and interpret its meaning in terms of this problem situation. The y-intercept is 300. In this situation, the y-intercept represents the amount in British pounds each student had initially, prior to exchanging any additional U.S. currency.. Analyzing Linear Functions 9 Is the slope and the rate of change the same thing? Where is the y-intercept located on the graph? Where does the graph of this function pass through the y-axis? Does the x-intercept have relevance in this problem situation? Why or why not?. Analyzing Linear Functions 93

38 Guiding Questions for Share Phase, Questions 4 and 5 What rules apply when rounding an amount of money? How many decimal places are used when rounding an answer that is an amount of money? What equation was used to determine the total pounds Dawson has if he exchanged an additional $50? How many steps were needed to solve the equation? What was the first step in the solution to the equation??4. Dawson would like to exchange $70 more. Jonathon thinks Dawson should have a total of Erin says he should have a total of , and Tre says he should have a total of 34. Who s correct? Who s reasoning is correct? Why are the other students not correct? Explain your reasoning. Jonathon f(d) = d f(d) = (70) f(d) = f(d) = Erin f(d) = d f(d) = (70) f(d) = f(d) = f(d) tre f (d ) = d f (d ) = (70) f (d ) = f (d ) = 34 The pound ( ) is made up of 100 pence (p), just like the dollar is made up of 100 cents. Erin is correct. Since Dawson is exchanging money, he must round to the hundredths place. Jonathon did not round his answer to reflect pence. Tre rounded the conversion rate and short-changed the actual amount of British pounds that Dawson should receive. 5. How many total pounds will Dawson have if he only exchanges an additional $50? Show your work. f(d ) d f(d ) (50) f(d ) If Dawson exchanges an additional $50, he will have a total of Chapter Graphs, Equations, and Inequalities

39 Problem 3 Using a graphing calculator and the function from the last problem, students are instructed on how to use the table and table set (TBLSET) features. Grouping Ask a student to read the instructions. Discuss as a class. Have students complete Questions 1 and with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Questions 1 and What function are you entering into the graphing calculator? Why do you want to start the table values at 100? Why do you want to set the delta table to 50? How can you adjust the delta table to view the values needed? PROBLEM 3 Using technology to Complete tables The exchange function is f (d ) = d. Throughout this lesson you used multiple representations and paper-and-pencil to answer questions. You can also use a graphing calculator to answer questions. Let s first explore how to use a graphing calculator to create a table of values for converting U.S. dollars to British pounds. You can use a graphing calculator to complete a table of values for a given function. Step 1: Press Y= Step : Enter the function. Press ENTER. Step 3: Press ND TBLSET (above WINDOW). TblStart is the starting data value for your table. Enter this value. Tbl (read delta table ) is the increment. This value tells the table what intervals to count by for the independent quantity. If Tbl = 1 then the values in your table would go up by 1s. If Tbl = -1, the values would go down by 1s. Enter the Tbl. Step 4: Press ND TABLE (above GRAPH). Use the up and down arrows to scroll through the data. 1. Use your graphing calculator and the TABLE feature to complete the table shown. U.S. Currency British Currency $ For this scenario, you will not exchange any currency less than $100. Set the TblStart value to 100. Analyze the given U.S. Currency dollar amounts and decide how to set the increments for DTbl Analyzing Linear Functions 9. Analyzing Linear Functions 95

40 . Were you able to complete the table using the TABLE feature? Why or why not? What adjustments, if any, can you make to complete the table? Answers will vary. The value (67, ) was not visible with the table settings in increments of 5. I can adjust the Tbl 5 1 to determine all the values. Problem 4 Using a graphing calculator and the function from the last problem, students are instructed on how to use the value and intersect features. Grouping Ask a student to read the instructions. Discuss as a class. Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. PROBLEM 4 Using technology to Analyze graphs If you get an error message, go back and adjust your WINDOW. ERR:INVALID 1: Quit : Goto There are several graphing calculator strategies you can use to analyze graphs to answer questions. Let s first explore the value feature. This feature works well when you are given an independent value and want to determine the corresponding dependent value. You can use the value feature on a graphing calculator to determine an exact data value on a graph. Step 1: Press Y=. Enter your function. Step : Press WINDOW. Set appropriate values for your function. Then press GRAPH. Step 3: Press ND and then CALC. Select 1:value.Press ENTER. Then type the given independent value next to X= and press ENTER. The cursor moves to the given independent value and the corresponding dependent value is displayed at the bottom of the screen. Be sure to double check that you typed in the correct function. Guiding Questions for Share Phase, Question 1 What function are you entering into the graphing calculator? Why are dimensions of the viewing window important? If the viewing window is not large enough, what will happen when you try to determine the value of the dependent variable? Will the value function on the graphing calculator help if you are given the dependent value and want to determine the independent value? Explain. Use the value feature to answer each question. 1. How many total British pounds will Amy have if she exchanges an additional: a. $375? Amy will have a total of if she exchanges an additional $375. b. $650? Amy will have a total of if she exchanges an additional $650. c. $000 Amy will have a total of if she exchanges an additional $ Chapter Graphs, Equations, and Inequalities

41 Guiding Questions for Share Phase, Questions and 3 Is there more than one correct answer? When would the use of the value function on the graphing calculator be helpful? Would the value function on the graphing calculator work if the function was a quadratic function? Grouping Ask a student to read the graphing calculator instructions. Discuss as a class.. How can you verify that each solution is correct? Answers will vary. I can use the function and estimate each value before I use the graphing value feature. I can double check to make sure I typed in the function correctly. 3. What are the advantages and limitations of using the value feature? Answers will vary. Using the value feature, I can easily determine the output value for any input value. A limitation is that I get an error message if I input an x-value that is not within the starting and ending values I specified. Let s now explore the intersect feature of CALC. You can use this feature to determine an independent value when given a dependent value. Suppose you know that Jorge has a total of You can first write this as f(d) d and y Then graph each equation, calculate the intersection point, and determine the additional amount of U.S. currency that Jorge exchanged. You can use the intersect feature to determine an independent value when given a dependent value. Step 1: Press Y5. Enter the two equations, one next to Y 1 5 and one next to Y 5. Step : Press WINDOW. Set appropriate bounds so you can see the intersection of the two equations. Then press GRAPH. Step 3: Press ND CALC and then select 5:intersect. The cursor should appear somewhere on one of the graphs, and at the bottom of the screen you will see First curve? Press ENTER. The cursor should then move to somewhere on the other graph, and you will see Second curve? Press ENTER. You will see Guess? at the bottom of the screen. Move the cursor to where you think the intersection point is and Press ENTER. The intersection point will appear. You can use your arrow keys to scroll to different features.. Analyzing Linear Functions 9. Analyzing Linear Functions 97

42 Grouping Ask a student to read the instructions before Question 4. Discuss as a class. Have students complete Questions 4 through 7 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Questions 4 through 7 What two functions are you entering into the graphing calculator? Are sets of parenthesis needed when entering either of the two functions? What are the upper and lower bounds needed to view a complete graph? If the viewing window is not large enough, what will happen when you try to determine the point of intersection? Is there more than one correct answer? When would the use of the intersect function on the graphing calculator be helpful? Would the intersect function on the graphing calculator work if one function was linear and the second function was quadratic? Problem 5 Use the intersect feature to answer each question. 4. How many additional U.S. dollars did Jorge exchange if he has a total of: a ? If Jorge has a total of 75.35, he exchanged an additional $ b ? If Jorge has a total of , he exchanged an additional $ How can you verify that each solution is correct? Answers may vary. I can use the function and estimate each value before I use the graphing value feature. I can double check to make sure I typed in the function correctly. 6. What are the advantages and limitations of the intersect feature? Answers will vary. I can easily determine a close approximation instead of trying to figure it out by inspection. A limitation is that the accuracy of the value depends on how well I guess at the point of intersection. 7. Do you think you could use each of the graphing calculator strategies discussed in this lesson with any function, not just linear functions? Answers will vary. I think I could use the strategies for many functions, but it may not be easy for functions that have more complicated structures. PROBLEM 5 gr aphing Calculator Practice Use a graphing calculator to evaluate each function. Explain the strategy you used. 1. f(x) x a. f(3.5) f(3.5) b. f(16.37) f(16.37) c. f(50.1) f(50.1) The graphing calculator is used to evaluate functions, sketch graphs, and determine the independent value given the dependent value. Grouping Have students complete Questions 1 through 4 with a partner. Then share the responses as a class. 98 Chapter Graphs, Equations, and Inequalities

43 Guiding Questions for Share Phase, Question 1 What screen on the graphing calculator did you use to evaluate the function? Is there a difference between using the negative key and using the subtraction key on the graphing calculator? Explain. When is it appropriate to use the negative key on the graphing calculator? When is it appropriate to use the subtraction key on the graphing calculator?. x x Use a graphing calculator and the intersect feature to determine each independent value. Then sketch the graphs on the coordinate plane provided. a. f(x) x 0 when f(x) y Be careful to use the negative key and the subtraction key properly. Also, remember to use parentheses when entering fractions. Guiding Questions for Share Phase, Questions and 3 Is there a difference between using the negative key and using the subtraction key on the graphing calculator? Explain. When is it appropriate to use the negative key on the graphing calculator? When is it appropriate to use the subtraction key on the graphing calculator? What screen on the graphing calculator did you use to complete the table of values? Is 3.38 the independent value or the dependent value in this function? Explain. What is the slope or rate of change of this equation? What is the y-intercept of this equation? x b. 1 4 x x Graphically speaking, where is the solution to this system of equations? Does this equation have more than one solution? Explain. y 0 10 Is 16 4 the independent value or the dependent value in this function? Explain. 5 What is the slope or rate of change of this equation? What is the y-intercept of this equation? Does this equation have more than one solution? Explain. Graphically speaking, where is the solution to this system of equations? x x. Analyzing Linear Functions 9. Analyzing Linear Functions 99

44 4. Use the word box to complete each sentence, and then explain your reasoning. always sometimes never If I am using a graphing calculator and I am given a dependent value and need to calculate an independent value, a. I can always use a table to determine an approximate value. I can adjust the Tbl. b. I can sometimes use a table to calculate an exact value. I might not be able adjust the Tbl in order for the table to show the exact dependent value that I was given. c. I can always use a graph to determine an approximate value. I can always graph two functions and approximate the intersection point. d. I can always use a graph to calculate an exact value. I can always graph two functions and determine the exact intersection point. e. I can always use a function to determine an approximate value. I can always use a guess and check strategy. f. I can sometimes use a function to calculate an exact value. I may not know the algebraic steps to solve the function. At this point, I only know how to solve linear equations. Be prepared to share your solutions and methods. 100 Chapter Graphs, Equations, and Inequalities

45 Check for Students Understanding Use the graph to answer each question. y Amusement Park Cost (0, 5.5) (1, 6.75) (, 8.5) (10, 0.5) (9, 18.75) (8, 17.5) (7, 15.75) (6, 14.5) (5, 1.75) (4, 11.5) (3, 9.75) Number of Rides 8 9 x 1. Complete a table of values. x (rides) f(x) (dollars) Analyzing Linear Functions 100A

46 . Describe a scenario to fit the graph. An amusement park charges a $5.5 admission fee and $1.50 for each ride. 3. Describe the independent variable with respect to the scenario. The independent variable is the number of rides. 4. Describe the dependent variable with respect to the scenario. The dependent variable is the total cost of going to the amusement park and riding. 5. Should the scenario be associated with a continuous graph or a discrete graph? Why? The scenario should be associated with a discrete graph because you cannot go on a fraction of a ride. 6. Is the relation a function? Explain your reasoning. The relation is a function because for every input there is exactly one output. 7. Write a function f(x) to represent the total cost of going to the amusement park as a function of the number of rides. f(x) x 8. Use your function to calculate the number of rides if the total amount of money spent in the amusement park is $3.5. f(x) x x x x 5 18 The number of rides you can go on for a total cost of $3.5 is 18 rides. 100B Chapter Graphs, Equations, and Inequalities

47 Scouting for Prizes! Modeling Linear Inequalities.3 Learning Goals Key Term In this lesson, you will: Write and solve inequalities. Analyze a graph on a coordinate plane to solve problems involving inequalities. Interpret how a negative rate affects how to solve an inequality. solve an inequality Essential Ideas Multiple representations such as tables, graphs, and inequalities are used to model linear situations. Solutions to linear inequalities are determined both graphically and algebraically. 3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. A-REI Reasoning with Equations and Inequalities Mathematics Common Core Standards A-CED Creating Equations Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve equations and inequalities in one variable 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Represent and solve equations and inequalities graphically 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 101A

48 N-Q Quantities Reason quantitatively and use units to solve problems. 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Overview This lesson is similar to the previous lesson but extends to include inequalities. Connections between the representations of linear inequalities on a coordinate plane and number line are made. Inequalities including positive and negative slopes are used in this lesson. Students will write functions to describe scenarios, analyze the functions, and graph the solutions. Solutions are determined both algebraically and graphically. Students will explore solving an inequality with a negative rate of change, which affects the sign of the inequality. 101B Chapter Graphs, Equations, and Inequalities

49 Warm Up Graph each inequality on the number line shown. 1. x $ 6. x, # x, Modeling Linear Inequalities 101C

50 101D Chapter Graphs, Equations, and Inequalities

51 Scouting for Prizes! Modeling linear inequalities.3 learning goals In this lesson, you will: Write and solve inequalities. Analyze a graph on a coordinate plane to solve problems involving inequalities. Inte rpret how a negative rate affects how to solve an inequality. KEy term solve an inequality couting began in 1907 by a Lieutenant General in the British Army, Robert Baden- S Powell, as a way to teach young men and women about different outdoor activities and survival techniques. While he was a military officer, Baden-Powell taught his soldiers how to survive in the wilderness and spent much time on scouting missions in enemy territory. He became a national hero during this time which helped fuel the sales of a book he had written, Aids to Scouting. When he returned home many people wanted him to rewrite his book for boys. While his rewritten book, Scouting for Boys, contained many of the same ideas about outdoor living, he left out the military aspects of his first book. Boys immediately began forming their own Scout patrols and wrote to Baden-Powell asking for his assistance. The Scouting movement has been growing and changing ever since. Do you think wilderness survival skills are necessary today? If yes, why do you think we still need these skills? If no, why do you think people still learn them if they are unnecessary?.3 Modeling Linear Inequalities 101

52 Problem 1 A scenario is given which is represented by an inequality with a positive slope. Students will write and analyze a function that describes the scenario. The analysis includes identifying the independent and dependent quantities including their units of measure, the rate of change, and the intercepts. A graph of the function is shown on a coordinate plane and the connection of the solution to a number line is made. Grouping Ask a student to read the scenario. Discuss as a class. Have students complete Questions 1 and with a partner. Then share the responses as a class. PROBLEM 1 Popcorn Pays Off Alan s camping troop is selling popcorn to earn money for an upcoming camping trip. Each camper starts with a credit of $5 toward his sales, and each box of popcorn sells for $3.75. Alan can also earn bonus prizes depending on how much popcorn he sells. The table shows the different prizes for each of the different sales levels. Each troop member can choose any one of the prizes at or below the sales level. Sales (dollars) Gift Cards ( of each value) $50 $10 $350 $15 $450 $0 Bonus Prizes $600 Cyclone Sprayer $650 $30 $850 $40 $1100 $55 $1300 $75 $1500 Choose your prize! $1800 $110 $300 $150 $500 6% toward college scholarship 10 Chapter Graphs, Equations, and Inequalities

53 Guiding Questions for Share Phase, Questions 1 and Under what conditions will Alan qualify for a bonus prize? If Alan s sales are less than $50, does qualify for a gift card? If Alan s sales are equal to $300, does he qualify for a gift card? What function family is the function associated with? Is the function written in the form of f(x) 5 ax 1 b? What is a? What is b? Is the slope negative or positive? 1. Write a function, f(b), to show Alan s total sales as a function of the number of boxes of popcorn he sells. f(b) b 1 5. Analyze the function you wrote. a. Identify the independent and dependent quantities and their units. The independent quantity is the number of boxes sold, and the dependent quantity is the total sales, in dollars. b. What is the rate of change and what does it represent in this problem situation? The rate of change is $3.75. This represents the cost of each box of popcorn. c. What is the y-intercept and what does it represent in this problem situation? The y-intercept is 5. This shows that every troop member starts with a $5 credit toward the total sales. How did you represent the $5 credit in your function?.3 Modeling Linear Inequalities 10.3 Modeling Linear Inequalities 103

54 Grouping Ask a student to read the information. Discuss the worked example as a class. Have students complete Questions 3 and 4 with a partner. Then share the responses as a class. Now, let s analyze your function represented on a graph. The graph shown represents the change in the total sales as a function of boxes sold. The oval and box represent the total sales at specific intervals. Total Sales ($) y f(b) b Number of Boxes Sold (40, 1600) x The box represents all the numbers of boxes sold, b, that would earn Alan $1600 or less. When f (b) 1600 then b 40. The oval represents all the numbers of boxes sold, b, that would earn Alan more than $1600. When f (b) > 1600, then b > Number of Boxes Sold The point at (40, 1600) means that at 40 boxes sold, the total sales is equal to $1600. This is represented on the number line as a closed point at 40. When f (b) = 1600, then b = Number of Boxes Sold Number of Boxes Sold 104 Chapter Graphs, Equations, and Inequalities

55 Guiding Questions for Share Phase, Questions 3 and 4 Under what condition is an open circle used on a number line? Under what condition is a closed circle used on a number line? What inequality sign is associated with the phrase at least? What inequality sign is associated with the phrase less than? What sign is associated with the word exactly? 3. Explain the difference between the open and closed circles on the number lines. The open circle means that the point is not included in the solution. The closed circle means that the point is included in the solution. 4. Use the graph to answer each question. Write an equation or inequality statement for each. a. How many boxes would Alan have to sell to earn at least $95? Alan would have to sell at least 40 boxes. b $ 40 b. How many boxes would Alan have to sell to earn less than $050? Alan would have to sell fewer than 540 boxes. b, 540 c. How many boxes would Alan have to sell to earn exactly $700? Alan would have to sell exactly 180 boxes. b How dœs determining the intersection point help you determine your answers?.3 Modeling Linear Inequalities 10.3 Modeling Linear Inequalities 105

56 Problem The solution set of an inequality with a positive slope is defined and determined using algebra. An example is provided and students focus on the similarities between algebraically solving an equation and solving an inequality. Grouping Ask a student to read the information and definition. Discuss the worked example and complete Question 1 as a class. Have students complete Question with a partner. Then share the responses as a class. PROBLEM What s your Strategy your Algebraic Strategy? Another way to determine the solution set of an inequality is to solve it algebraically. To solve an inequality means to determine the values of the variable that make the inequality true. The objective when solving an inequality is similar to the objective when solving an equation: You want to isolate the variable on one side of the inequality symbol. In order to earn two $55 gift cards, Alan s total sales, f(b), needs to be at least $1100. You can set up an inequality and solve it to determine the number of boxes Alan needs to sell. f(b) $ b 1 5 $ 1100 Solve the inequality in the same way you would solve an equation. 3.75b 1 5 $ b $ b $ b 3.75 $ b $ Alan needs to sell at least 87 boxes of popcorn to earn two $55 gift cards. Guiding Questions for Discuss Phase, Question 1 What was the first step in the solution to the inequality? Is it possible to sell boxes of popcorn? If Alan sold 86 boxes would he earn two $55 gift cards? 1. Why was the answer rounded to 87? Alan can t sell partial boxes of popcorn. If he sold 86 boxes, that would not be enough to earn at least $ Write and solve an inequality for each. Show your work. a. What is the greatest number of boxes Alan could sell and still not have enough to earn the Cyclone Sprayer? 3.75b 1 5, b 1 5 5, b, b 3.75, b, Alan could sell at most 153 boxes of popcorn and still not have enough to earn the Cyclone Sprayer. Guiding Questions for Share Phase, Question How much in sales does Alan need to qualify for the Cyclone Sprayer? What inequality sign is used to solve this problem? Why? What was the first step in the solution to the inequality? If Alan sold 153 boxes, what would be his total sales? If Alan sold 154 boxes, what would be his total sales? How much in sales does Alan need to qualify for choosing his own prize? 106 Chapter Graphs, Equations, and Inequalities

57 Guiding Questions for Share Phase, Question What inequality sign is used to solve this problem? Why? What was the first step in the solution to the inequality? If Alan sold 393 boxes, what would be his total sales? If Alan sold 394 boxes, what would be his total sales? b. At least how many boxes would Alan have to sell to be able to choose his own prize? 3.75b 1 5 $ b $ b $ b 3.75 $ b $ Alan would need to sell at least 394 boxes to be able to choose his own prize. PROBLEM 3 reversing the Sign Problem 3 A scenario is given which is represented by an inequality with a negative slope. Students will write and analyze a function that describes the scenario. The analysis includes identifying the independent and dependent quantities including their units of measure, the rate of change, and the intercepts. A graph of the function is given on a coordinate plane and students will use the graph to determine unknowns. They verify the solution algebraically and compare the graphic solution to the algebraic solution to conclude they are one in the same. Next, students complete a table of values and describe the change of the inequality sign when multiplication or division of a negative number is part of the solution process. Grouping Have students complete Questions 1 and with a partner. Then share the responses as a class. Alan s camping troop hikes down from their campsite at an elevation of 4800 feet to the bottom of the mountain. They hike down at a rate of 0 feet per minute. 1. Write a function, h(m), to show the troop s elevation as a function of time in minutes. h(m) 5 0m Analyze the function. a. Identify the independent and dependent quantities and their units. The independent quantity is the number of minutes hiked, and the dependent quantity is the elevation in feet. b. Identify the rate of change and explain what it means in terms of this problem situation. The rate of change is 0. This represents a decrease of 0 feet every minute. c. Identify the y-intercept and explain what it means in terms of this problem situation. The y-intercept is This shows that the troop started their hike at an elevation of 4800 feet. d. What is the x-intercept and explain what it means in terms of this problem situation? 0 5 0m m m 0m 40 5 m The x-intercept is (40, 0). The hikers will be at the bottom of the mountain in 40 minutes, or 4 hours..3 Modeling Linear Inequalities 10 Guiding Questions for Share Phase, Questions 1 and What sign is associated with hiking down a mountain? What sign is associated with hiking up a mountain? What function family is associated with this situation? Is the function written in the form of f(x) 5 ax 1 b? What is a? What is b? Is the slope negative or positive? Would the graph of this function pass through the origin? Why or why not? What equation is used to determine the x-intercept?.3 Modeling Linear Inequalities 107

58 Grouping Have students complete Questions 3 through 6 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Questions 3 through 6 Should the oval be drawn above or below the line y 5 300? Why? What inequality is used to verify the solution? What inequality sign is used? Why? What is the first step in the solution process? When both sides of the inequality are divided by 80, what happens to the inequality sign? Why does this happen? When a negative value is added to both sides of an inequality, does this cause a change in the inequality sign? Why or why not? When both sides of an inequality are divided by a negative number, does this cause a change in the inequality sign? Why or why not? When both sides of an inequality are multiplied by a negative number, does this cause a change in the inequality sign? Why or why not? 3. Label the function on the coordinate plane. Campsite Elevation (feet) y h(m) 5 0m Time (minutes) x y Use the graph to determine how many minutes passed if the troop is below 300 feet. Draw an oval on the graph to represent this part of the function and write the corresponding inequality statement. More than 80 minutes has passed if the troop is below 300 feet. m Write and solve an inequality to verify the solution set you interpreted from the graph. 0m , 300 0m , m, m 0, 1600m 0m m Compare and contrast your solution sets using the graph and the function. What do you notice? The solution sets are the same. 108 Chapter Graphs, Equations, and Inequalities

59 Grouping Have students complete Question 7 with a partner. Then share the responses as a class. 7. Complete the table by writing the corresponding inequality statement that represents the number of minutes for each height. h(m) m Guiding Questions for Share Phase, Question 7 When solving an inequality, when is it not necessary to reverse the inequality sign? How is solving an inequality the same as and different from solving an equation? h(m). 300 m, 80 h(m) $ 300 m # 80 h(m) m 5 80 h(m), 300 m. 80 h(m) # 300 m $ 80 a. Compare each row in the table shown. What do you notice about the inequality signs? The inequality signs are reversed in each row. b. Explain your answer from part (a). Use what you know about solving inequalities when you have to multiply or divide by a negative number. The function included a negative rate so when I divide by that negative rate to solve the inequality, the sign reversed. To solve the inequalities for m, I have to divide both sides of the inequalities by a negative number. This gives the solutions I determined in the table. The signs must be reversed..3 Modeling Linear Inequalities 10.3 Modeling Linear Inequalities 109

60 Talk the Talk Students will explain the differences between solving an inequality with a positive slope and solving an inequality with a negative slope. They also solve 3 inequalities and graph the solution on number lines. Grouping Have students complete Questions 1 and with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Questions 1 and Which function contained a positive rate of change? Which function contained a negative rate of change? If a function contains a negative rate of change, will the solution process always contain a step requiring the reversal of the inequality sign? If a function contains a positive rate of change, will the solution process always contain the same inequality sign? talk the talk 1. Explain the differences when you solved inequalities involving the function f(b) b 1 5 and the function h(m) 5 0m The function f(b) contained a positive rate of change. When I solved inequalities using f(b), the sign of the inequality did not change because I did not multiply or divide by a negative rate. The function h(m) contained a negative rate of change. When I solved inequalities using h(m), the sign of the inequality reversed because I divided by a negative rate.. Solve each inequality and then graph the solution on the number line. a. 3 x $ 7 ( 3 ) ( 3 ) x $ ( 3 ) 7 x # 1 b x x 9. x 9, x x. 9 c. (x 1 6), 10 x 1 1, 10 x 1 1 1, 10 1 x, x, x, 1 Be prepared to share your solutions and methods Chapter Graphs, Equations, and Inequalities

61 Check for Students Understanding Solve each for x. 1. 7x 4 5 x x 4. x x x x x x 1 x 5 x 1 x 1 0 7x 1 x. x 1 x 1 0 5x 5 0 5x. 0 x 5 4 x, 4 3. Is problem 1 and problem the same problem? Explain your reasoning. Problem 1 and problem are not the same problem. Problem 1 is an equation and problem is an inequality. 4. Is the solution to problem 1 and problem the same? Explain your reasoning. The solutions to problem 1 and problem are not the same. The solution to problem 1 is a single value, 4. The solution to problem is an infinite number of values all less than Is the strategy to solve problem 1 and problem similar? Explain your reasoning. The strategy used to solve problem 1 and problem are similar. The first step in both problems is to add 4 to both sides of the equality or inequality, the second step is to add x to both sides, and the third step is to divide both sides by Did the equality sign in problem 1 change in any step of the solution? Explain your reasoning. No. The equality sign did not change in any step of the solution. Every step of the equation contains an equal sign. 7. Did the inequality sign in problem change in any step of the solution? Explain your reasoning. Yes. The inequality sign changed in the last step of the solution. The inequality began as greater than however it was necessary to change the sign to greater than in the last step of the solution because we divided both sides of the inequality by a negative number..3 Modeling Linear Inequalities 110A

62 110B Chapter Graphs, Equations, and Inequalities

63 We re Shipping Out! Solving and Graphing Compound Inequalities.4 Learning Goals Key Terms In this lesson, you will: Write simple and compound inequalities. Graph compound inequalities. Solve compound inequalities. compound inequality solution of a compound inequality conjunction disjunction Essential Ideas A compound inequality is an inequality that is formed by the union, or, or the intersection, and of two simple inequalities. Compound inequalities are written in compact form. The solution of a compound inequality (conjunction) written in the form a, x, b, where a and b are any real numbers, is the part or parts of the solutions that satisfy both of the inequalities. The solution of a compound inequality (disjunction) written in the form x, a or x. b, where a and b are any real numbers, is the part or parts of the solutions that satisfy either of the inequalities. Mathematics Common Core Standards A-CED Creating Equations Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-REI Reasoning with Equations and Inequalities Solve equations and inequalities in one variable 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 111A

64 Overview Compound inequalities are defined in this lesson. In the first problem students are given a scenario that is represented by compound inequalities. They will answer questions related to the scenario and express the compound inequalities in compact form. In the second problem students are given three different scenarios. Students express the inequalities using symbols, graph the inequalities, and solve the inequalities. In the third problem students represent the solutions to compound inequalities written in compact form on number lines and show the algebraic solutions. The terms conjunction and disjunction are defined. 111B Chapter Graphs, Equations, and Inequalities

65 Warm Up Explain the error(s) in each graph. 1. x # The circle should be closed.. x $ The circle should be closed and the arrow should be reversed # x, The circle on the right should be open..4 Solving and Graphing Compound Inequalities 111C

66 111D Chapter Graphs, Equations, and Inequalities

67 We re Shipping Out! Solving and graphing Compound inequalities learning goals KEy terms.4 In this lesson, you will: Write simple and compound inequalities. Graph compound inequalities. Solve compound inequalities. compound inequality solution of a compound inequality conjunction disjunction How many different ways do you think water exists? You may instantly think of water in a liquid state like you see in raindrops, or in lakes, ponds, or oceans. However, you probably also know that water can be a solid like hail, or ice cubes; or as a gas as in the humidity you may feel on a hot summer day, or the steam you see. What factors do you think play a role in the way water exists? Can you think of other things that can take the form of a solid, liquid, and gas?.4 Solving and Graphing Compound Inequalities 111

68 Problem 1 A scenario is given which is represented by a compound inequality. Students will answer questions related to the different levels of the cost for shipping goods. They use the phrases greater than or equal to or less than or equal to to describe the costs related to the different shipping costs. The term compound inequality is defined and students express each category of shipping costs as a compound inequality. Examples of writing compound inequalities in compact form are given and students use these examples to write all of the compound inequalities relevant to the scenario in compact form. PROBLEM 1 goodsportsbuys.com GoodSportsBuys.com is an online store that offers discounts on sports equipment to high school athletes. When customers buy items from the site, they must pay the cost of the items as well as a shipping fee. At GoodSportsBuys.com, a shipping fee is added to each order based on the total cost of all the items purchased. This table provides the shipping fee categories for GoodSportsBuys.com. Total Cost of Items Shipping Fee $0.01 up to and including $0 $6.50 More than $0 up to and including $50 $9.00 Between $50 and $75 $11.00 From $75 up to, but not including, $100 $1.5 $100 or more $ What is the least amount a customer can spend on items and pay $6.50 for shipping? The least amount is $0.01. Grouping Ask a student to read the scenario. Discuss as a class. Have students complete Questions 1 through 5 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Questions 1 through 5 Why do the shipping fees increase as the total cost of the items increase? What other methods do companies use to determine what to charge customers for shipping?. What is the greatest amount a customer can spend on items and pay $6.50 for shipping? The greatest amount is $ What is the shipping fee if Sarah spends exactly $75.00 on items? Explain your reasoning. If she buys items that cost exactly $75, Sarah will pay $1.5, because $75 is not between $50 and $75.? He is correct, because the shipping cost for purchases of $ or more is $ Harvey says he will spend $13.10 on shipping fees if he spends exactly $100 on items. Is he correct? Explain your reasoning. What is the shipping fee associated with a baseball catcher s mitt that costs $34.99? If someone paid $11 in shipping fees, what was the total cost of the items he purchased, not including the shipping fee? How is Between $50 and $75 expressed using symbols? What phrases were used in the table to help determine what shipping fees to apply to the boundary values in the table? 11 Chapter Graphs, Equations, and Inequalities

69 Grouping Ask a student to read the definition. Discuss as a class. Have students complete Questions 6 and 7 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Questions 6 and 7 What does the term union mean? How is the term union related to the term or? What does the term intersection mean? How is the term intersection related to the term and? What makes a mathematical statement a compound inequality? 5. Consider the table of shipping costs to complete each statement using the phrase greater than, less than, greater than or equal to, or less than or equal to. a. You will pay $6.50 in shipping fees if you spend: greater than or equal to $0.01, and less than or equal to $0.00 on items b. You will pay $9.00 in shipping fees if you spend: greater than $0.00, and less than or equal to $50.00 on items c. You will pay $11.00 in shipping fees if you spend: greater than $50.00, and less than $75.00 on items d. You will pay $1.5 in shipping fees if you spend: greater than or equal to $75.00, and less than $ on items e. You will pay $13.10 in shipping fees if you spend: greater than or equal to $ on items A compound inequality is an inequality that is formed by the union, or, or the intersection, and, of two simple inequalities. 6. You can use inequalities to represent the various shipping fee categories at GoodSportsBuys.com. If you let x represent the total cost of items purchased, you can write an inequality to represent each shipping fee category. Complete each inequality using an inequality symbol. a. $6.50 shipping fees: x $ $0.01 and x # $0 b. $9.00 shipping fees: x. $0 and x # $50 c. $11.00 shipping fees: x. $50 and x, $75 d. $1.5 shipping fees: x $ $75 and x, $100 e. $13.10 shipping fees: x $ $ Identify the inequalities in Question 6 that are compound inequalities. Parts (a) through (d) are compound inequalities..4 Solving and Graphing Compound Inequalities 11.4 Solving and Graphing Compound Inequalities 113

70 Grouping Ask a student to read the information. Discuss the worked examples and complete Question 8 as a class. Let s consider two examples of compound inequalities. Guiding Questions for Discuss Phase, Question 8 What does it mean to write an inequality in compact form? Why is it only possible for compound inequalities containing and to be written in compact form? How did you determine whether or not the inequality symbol should include the equals sign? x. and x # 7 This inequality is read as all numbers greater than and less than or equal to 7. This inequality can also be written in the compact form of, x # 7. x # 4 or x. This inequality is read as all numbers less than or equal to 4 or greater than. Only compound inequalities containing and can be written in compact form. 8. Write the compound inequalities from Question 6 using the compact form. a. $6.50 shipping fees: $0.01 # x # $0 b. $9.00 shipping fees: $0, x # $50 c. $11.00 shipping fees: $50, x, $75 d. $1.5 shipping fees: $75 # x, $ Chapter Graphs, Equations, and Inequalities

71 Problem Three different scenarios are given which can be represented using compound inequalities. Students will write the compound inequalities to represent each scenario and graph them using number lines. Questions focus on solving the inequalities. Grouping Have students complete Questions 1 and with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Questions 1 and At what temperature, in degrees Fahrenheit, is water as a liquid? At what temperature, in degrees Fahrenheit, is water as a gas? At what temperature, in degrees Fahrenheit, is water as a solid? Does your graph contain open or closed circles? Why? Why does it make sense to draw a circle to represent all the possible locations of Luke s and Logan s home? What is the distance between the two circles? What is the diameter of each circle? Where are the possible locations for Luke s house and Logan s house if the houses are one mile apart? PROBLEM More than One Solution 1. Water becomes non-liquid when it is 3 F or below, or when it is at least 1 F. a. Represent this information on a number line b. Write a compound inequality to represent the same information. Define your variable. Let w represent the water in a non-liquid state. w # 3 or w $ 1. Luke and Logan play for the same baseball team. They practice at the Lions Park baseball field. Luke lives 3 miles from the field, and Logan lives miles from the field. a. First, plot a point to represent the location of the Lions Park baseball field. b. Next, use your point that represents Lions Park, and draw a circle to represent all the possible places Luke could live. c. Finally, use your point that represents Lions Park, and draw another circle to represent all the possible places Logan could live. miles 3 miles Possible location of Logan s home Possible location of Luke s home d. What is the shortest distance, d, that could separate their homes? The shortest distance between the boys two homes is 1 mile. e. What is the longest distance, d, that could separate their homes? The longest distance between the boys two homes is 5 miles. Where are the possible locations for Luke s house and Logan s house if the houses are five miles apart? Where are the possible locations for Luke s house and Logan s house if the houses are between one and five miles apart? How can the compound inequality be written differently? Does your graph contain open or closed circles? Why? field.4 Solving and Graphing Compound Inequalities 11.4 Solving and Graphing Compound Inequalities 115

72 Grouping Have students complete Question 3 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Question 3 Is it better to have a higher miles per gallon ratio or a lower miles per gallon ratio? Why? Why do cars have a lower miles per gallon ratio in cities rather than on highways? What does 14m represent? Why must you divide by 14 in 3 places when solving the compound inequality when it is written in compact form? Do you prefer to solve compound inequalities containing and as two simple inequalities or as one compound inequality written in compact form? Why? f. Write a compound inequality to represent all the possible distances that could separate their homes. 1 # d # 5 g. Represent the solution on a number line Jodi bought a new car with a 14-gallon gas tank. Around town she is able to drive 336 miles on one tank of gas. On her first trip traveling on highways, she drove 448 miles on one tank of gas. What is her average miles per gallon around town? What is her average miles per gallon on highways? a. Write a compound inequality that represents how many miles Jodi can drive on a tank of gas. Let m represents the number of miles per gallon of gas. 336 # 14m # 448 b. Rewrite the compound inequality as two simple inequalities separated by either and or or. 336 # 14m and 14m # 448 c. Solve each simple inequality. 336 # 14m and 14m # # 14m 14m # # m m # 3 4 # m and m # 3 d. Go back to the compound inequality you wrote in Question 3, part (a). How can you solve the compound inequality without rewriting it as two simple inequalities? Solve the compound inequality. I can divide all parts of the compound inequality by # 14m # # 14m 14 # # m # 3 e. Compare the solution you calculated in Question 3, part (c) with the solution you calculated in Question 3, part (d). What do you notice? The solution to part (d) is the same as the solution to part (c). The only difference is that part (d) is written in compact form. f. Explain your solution in terms of the problem situation. Jodi will be able to drive between 4 and 3 miles, inclusive, per one gallon of gas in her new car. 116 Chapter Graphs, Equations, and Inequalities

73 Problem 3 The solutions of compound inequalities for both conjunctions and disjunctions are defined. Students begin by classifying the inequalities in the previous problem as either conjunctions or disjunctions. Worked examples are provided for graphing both a conjunction and a disjunction on number lines. Students will then consider how the solution set of each example may be affected if the compound inequality was changed from a conjunction to a disjunction and a disjunction to a conjunction. Next, several conjunctions and disjunctions are given and students will graph each inequality on a number line. Students then solve and graph compound inequalities written in compact form. Grouping Ask a student to read the definitions. Complete Question 1 and discuss the worked examples as a class. g. Represent the solution on a number line. Describe the shaded region in terms of the problem situation The shaded region represents the possible miles Jodi s car will get on one gallon of gas. PROBLEM 3 Solving Compound inequalities Remember, a compound inequality is an inequality that is formed by the union, or, or the intersection, and, of two simple inequalities. The solution of a compound inequality in the form a, x, b, where a and b are any real numbers, is the part or parts of the solutions that satisfy both of the inequalities. This type of compound inequality is called a conjunction. The solution of a compound inequality in the form x, a or x. b, where a and b are any real numbers, is the part or parts of the solution that satisfy either inequality. This type of compound inequality is called a disjunction. 1. Classify each solution to all the questions in Problem as either a conjunction or disjunction. The solution to Question 1 is a disjunction while the solutions to Questions and 3 are conjunctions. Let s consider two examples for representing the solution of a compound inequality on a number line. The compound inequality shown involves and and is a conjunction. x # 1 and x. 3 Represent each part above the number line. x 1 x > Guiding Questions for Discuss Phase, Question 1 Does the term union apply to a conjunction or disjunction? Does the term intersection apply to a conjunction or disjunction? What is the difference between a conjunction and a disjunction? x 1 and x > 3 3 < x 1 The solution is the region that satisfies both inequalities. Graphically, the solution is the overlapping, or the intersection, of the separate inequalities..4 Solving and Graphing Compound Inequalities Solving and Graphing Compound Inequalities 117

74 Guiding Questions for Discuss Phase, Worked Examples How is the graph of a disjunction different from the graph of a conjunction? In the first worked example, why is the intersection of the two simple inequalities the solution to this compound inequality? In the first worked example, why is the open circle used on the left side and a closed circle used on the right side of the solution? Why is the union of the two simple inequalities the solution to this compound inequality? Do you prefer to list solutions to a compound inequality from its algebraic form or from its graphical form? Why? Grouping Have students complete Question 3 with a partner. Then share the responses as a class. The compound inequality shown involves or and is a disjunction.. Consider the two worked examples in a different way. a. If the compound inequality in the first worked example was changed to the disjunction, x # 1 or x. 3, how would the solution set change? Explain your reasoning. The solution set would be the set of all real numbers because the union of the individual inequalities would overlap. x # 1 x, or x. 1 Represent each part above the number line. x < x < or x > 1 The solution is the region that satisfies either inequality. Graphically, the solution is the union, or all the regions, of the separate inequalities. x b. If the compound inequality in the second worked example was changed to the conjunction, x, or x. 1, how would the solution set change? Explain your reasoning. There would be no solution because the graphs of the individual inequalities would not intersect. x, x Represent the solution to each compound inequality on the number line shown. Then, write the final solution that represents the graph. a. x, or x. 3 x > x, or x Chapter Graphs, Equations, and Inequalities

75 Guiding Questions for Share Phase, Question 3 How did you decide whether the solution was the intersection or union of the two graphs? What are three values that are considered solutions to the compound inequality? How did you determine these? For part (b), how would the solution be different if both circles were open circles? For part (d), why does it make sense that there are no solutions to this compound inequality? For part (g), why does it make sense that all real numbers are solutions to the compound inequality? Will the solutions to all compound inequalities containing or be two rays traveling in opposite directions? Why or why not? b. 1 $ x $ x 5 1 c. x, 0 or x, x, d. x. 1 and x, There is no solution. e. x, 3 and x , x, 3 f. x, and x, x, 1 Pay attention to whether the inequality uses and or or. g. x. 1 or x, All real numbers.4 Solving and Graphing Compound Inequalities 11.4 Solving and Graphing Compound Inequalities 119

76 Guiding Have students complete Question 4 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Question 4 How are the steps to solving a compound inequality algebraically different from the steps to solving a simple inequality algebraically? How can you verify that your solution is correct? Why was it necessary to reverse the inequality symbols in some of the problems? How could you rewrite the solution to the compact inequality statement using, or # symbols rather than. or $ symbols? To solve a compound inequality written in compact form, isolate the variable between the two inequality signs, and then graph the resulting statement. To solve an inequality involving or, simply solve each inequality separately, keeping the word or between them, and then graph the resulting statements. 4. Solve and graph each compound inequality showing the steps you performed. Then, write the final solution that represents the graph. a. 6, x 6 # 9 6, x 6 # , x # , x # b., x, 6, x, 6 1. x , x, c. 4 # 3x 1 1 # 1 4 # 3x 1 1 # # 3x # # 3x # $ 3x 3 $ $ x $ d. x 1 7, 10 or x x 1 7, 10 or x x 1 7 7, 10 7 or x x, 3 or x. 3 x, 3 or x, 3 x, 3 or x, x, Chapter Graphs, Equations, and Inequalities

77 e. 1 x or x, 3 1 x or x, 3 1 x or x () 1 x. ()1 or x. 3 x. x f x. 11 or x 4, x. 11 or x 4, x or x 4 1 4, x. 10 or x, 1 6x x Be prepared to share your solutions and methods..4 Solving and Graphing Compound Inequalities 1.4 Solving and Graphing Compound Inequalities 11

78 Check for Students Understanding Match each graph with the correct inequality. Graph Inequality 1. (C) A. x 1 8 # 1 and 3x (E) B. x 14 # 1 4 or x (D) C. x 14 # 1 4 and x (A) D. 1, 3x 1 6 or 5x 1 8 $ (B) E. 1, 3x 1 6 and 5x 1 8 $ 17 1 Chapter Graphs, Equations, and Inequalities

79 Play Ball! Absolute Value Equations and Inequalities.5 Learning Goals Key TermS In this lesson, you will: Understand and solve absolute values. Solve linear absolute value equations. Solve and graph linear absolute value inequalities on number lines. Graph linear absolute values and use the graph to determine solutions. opposites linear absolute value linear absolute value equation linear absolute value inequality equivalent compound inequalities Essential Ideas Opposite numbers are two numbers that are equal distance from zero on the number line. The absolute value of a number is the distance the number is from zero on a number line. A linear absolute value equation can be rewritten as a positive and a negative equation. Linear absolute value inequalities can be used to represent situations. Linear absolute value inequalities can be rewritten as equivalent compound inequalities. Mathematics Common Core Standards A-CED Creating Equations Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. A-REI Reasoning with Equations and Inequalities Solve equations and inequalities in one variable 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Represent and solve equations and inequalities graphically 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 13A

80 Overview This lesson begins with a review of absolute value including graphing numbers on a number line, determining the number of solutions, and using algebra to determine solutions. Scenarios are used to write two equations and then rewrite the two equations as one linear absolute value equation. An example shows how to solve linear absolute value equations by separating the equation into a positive and negative equation. Students will then solve linear absolute value equations and graph the solution. In the remaining problems, students extend their understanding of solving linear equations and consider linear absolute value inequalities. A scenario is given and students will complete a table of values, write a linear absolute value equation, and graph it on a coordinate plane. They then answer questions using the graph and connect those answers to the algebraic solution of the corresponding linear absolute value inequality. Finally, students will solve linear absolute value inequalities by rewriting them as equivalent compound inequalities. 13B Chapter Graphs, Equations, and Inequalities

81 Warm Up Answer each question. 1. What values are 6 units from the value 0 on the number line? The values 6 and 6 are 6 units from the value 0 on the number line.. What values are 3 units from the value 11 on the number line? The values 8 and 14 are 3 units from the value 11 on the number line. 3. What values are 8 units from the value 5 on the number line? The values 3 and 13 are 8 units from the value 5 on the number line. 4. What values are 9 units from the value on the number line? The values 11 and 7 are 9 units from the value on the number line. 5. What values are 7 units from the value 10 on the number line? The values 17 and 3 are 7 units from the value 10 on the number line. 6. What do Questions 1 through 5 and their responses have in common? All questions deal with a distance from a number. There are two answers to every question..5 Absolute Value Equations and Inequalities 13C

82 13D Chapter Graphs, Equations, and Inequalities

83 Play Ball! Absolute Value Equations and inequalities learning goals KEy terms.5 In this lesson, you will: Understand and solve absolute values. Solve linear absolute value equations. Solve and graph linear absolute value inequalities on number lines. Graph linear absolute values and use the graph to determine solutions. opposites absolute value linear absolute value equation linear absolute value inequality equivalent compound inequalities All games and sports have specific rules and regulations. There are rules about how many points each score is worth, what is in-bounds and what is out-ofbounds, and what is considered a penalty. These rules are usually obvious to anyone who watches a game. However, some of the regulations are not so obvious. For example, the National Hockey League created a rule that states that a blade of a hockey stick cannot be more than three inches or less than two inches in width at any point. In the National Football League, teams that wear black shoes must wear black shoelaces and teams that wear white shoes must wear white laces. In the National Basketball Association, the rim of the basket must be a circle exactly 18 inches in diameter. Most sports even have rules about how large the numbers on a player s jersey can be! Do you think all these rules and regulations are important? Does it really matter what color a player s shoelaces are? Why do you think professional sports have these rules, and how might the sport be different if these rules did not exist?.5 Absolute Value Equations and Inequalities 13

84 Problem 1 Opposites are described as two numbers that are equal distance from zero on the number line. The absolute value of a number is defined as the distance the number is from zero on a number line. Students will graph opposites on number lines and evaluate the absolute value of several numbers. Equations containing absolute values are given and students determine the number of solutions for each equation. They also evaluate algebraic expression containing absolute values. Grouping Ask a student to read the introduction. Discuss and complete Questions 1 through 3 as a class. Have students complete Questions 4 through 8 with a partner. Then share the responses as a class. Guiding Questions for Discuss Phase, Questions 1 through 3 Is there any visual relationship between and on the number line? What do all three graphs have in common? How are the graphs different? If the number line was folded vertically through the value of 0, would the two points lie on top of each other? Is this significant? Why? PROBLEM 1 Opposites Attract? Absolutely! 1. Analyze each pair of numbers and the corresponding graph. a. and b. and c. 1.5 and Describe the relationship between the two numbers. The numbers are opposites of each other. 3. What do you notice about the distance each point lies away from zero on each number line? For each set of opposite numbers, the distance between each number and zero is the same. Two numbers that are an equal distance, but are in different directions, from zero on the number line are called opposites. The absolute value of a number is its distance from zero on the number line. 4. Write each absolute value. a b. 3 5 Guiding Questions for Share Phase, Question 4 How far from 0 is on the number line? How far from 0 is on the number line? Is there such a thing as 0? 3 5 c What do you notice about each set of answers for Question 4? All answers are positive. The absolute values of numbers that are opposites of each other are the same. 3 How can you use each corresponding graph in Question 1 to verify your answers? 14 Chapter Graphs, Equations, and Inequalities

85 Guiding Questions for Share Phase, Questions 6 through 8 When evaluating the expression 3 8 is 3 subtracted from 8 or is 8 subtracted from 3? Is there a difference? Is it possible to subtract 8 from 3? How is it possible to subtract a larger number from a smaller number? Are the answers to Question 7 part (a) and part (b) the same? How many answers in Question 7 are negative? If x 5 5, can x also equal 5? Why or why not? If x 5 5, can x equal both 5 and 5? If x 5 5, can x equal both 5 and 5? 6. Determine the value of each. Show your work. a. 3 8 b. 3 8 c. 4(5) d. 4? 5 e. 1 f ? Determine the solution(s) to each equation a. x 5 5 There is only one solution. x 5 5 c. x 5 5 There are no solutions. Absolute values cannot be negative because they are a measure of distance from zero. b. x 5 5 There are two solutions. x 5 5 or x 5 5 d. x 5 0 There is one solution. x 5 0 Use the number line as a tool to think about each solution. 8. Analyze each equation containing an absolute value symbol in Question 7. What does the form of the equation tell you about the possible number of solutions? If the absolute value is equal to a positive number, then there are two possible solutions. If the absolute value is equal to a negative number, then there is no possible solution. If the absolute value is equal to zero, then there is only one possible solution..5 Absolute Value Equations and Inequalities 1.5 Absolute Value Equations and Inequalities 15

86 Problem Students will use the scenario to write and solve two equations to determine the lightest and heaviest baseball acceptable within the official baseball weight regulations. An example is provided that rewrites the two equations as one linear absolute value equation and shows the solution steps to a linear absolute value equation. Students then solve or evaluate different linear absolute value equations and explain errors in given solution processes. Grouping Ask a student to read the information. Discuss as a class. Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. Guiding Questions for Discuss Phase, Questions 1 through 3 How many different weights can a baseball have within the weight regulations? What is the target weight of a baseball? How is it used to solve the problem? What is the largest acceptable difference in weight? What is the smallest acceptable difference in weight? PROBLEM too heavy? too light? you re Out! The official rules of baseball state that all baseballs used during professional games must be within a specified range of weights. The baseball manufacturer sets the target weight of the balls at grams on its machines. The specified weight allows for a difference of 3.95 grams. This means the weight can be 3.95 grams greater than or less than the target weight. 1. Write an expression to represent the difference between a manufactured baseball s weight and the target weight. Use w to represent a manufactured baseball s weight. w Suppose the manufactured baseball has a weight that is greater than the target weight. a. Write an equation to represent the greatest acceptable difference in the weight of a baseball. w b. Solve your equation to determine the greatest acceptable weight of a baseball. w w w The greatest acceptable weight of a baseball is grams. 3. Suppose the manufactured baseball has a weight that is less than the target weight. a. Write an equation to represent the least acceptable difference in weight. w b. Solve your equation to determine the least acceptable weight of a baseball. w w w The least acceptable weight of a baseball is grams. What is done to the target weight to determine the largest acceptable difference in weight? What is done to the target weight to determine the smallest acceptable difference in weight? What is the difference between the largest acceptable difference in weight and the smallest acceptable difference in weight? 16 Chapter Graphs, Equations, and Inequalities

87 Grouping Ask a student to read the definition. Discuss the worked example as a class. Have students complete Questions 4 and 5 with a partner. Then share the responses as a class. The two equations you wrote can be represented by the linear absolute value equation w In order to solve any absolute value equation, recall the definition of absolute value. Consider this linear absolute value equation. a 5 6 There are two points that are 6 units away from zero on the number line: one to the right of zero, and one to the left of zero. 1(a) 5 6 or (a) 5 6 Now consider the case where a 5 x 1. a 5 6 or a 5 6 x If you know that a 5 6 can be written as two separate equations, you can rewrite any absolute value equation. 1(a) 5 6 or (a) 5 6 1(x 1) 5 6 or (x 1) 5 6 To solve each equation, would it be more efficient to distribute the negative or divide both sides of the equation by 1 first? 4. How do you know the expressions 1(a) and (a) represent opposite distances? When I add 1(a) and (a), I get a sum of Determine the solution(s) to the linear absolute value equation x Then check your answer. 1(x 1) 5 6 (x 1) 5 6 x x x or x x 5 7 x 5 5 The expressions +(x 1) and (x 1) are opposites..5 Absolute Value Equations and Inequalities 1.5 Absolute Value Equations and Inequalities 17

88 Grouping Have students complete Questions 6 through 8 with a partner. Then share the responses as a class Guiding Questions for Share Phase, Questions 6 through 8 How can the equation x be rewritten as two equations? How can the equation x be rewritten as two equations? How can the equation 3x be rewritten as two equations? How can the equation x be rewritten as two equations? Do all linear absolute value equations always have a solution? Why or why not? Do linear absolute value equations always have two solutions? Why or why not? When does a linear absolute value equation have one solution? Is it easier to solve the equation if you isolate the absolute value part of the equation first? Why or why not? 6. Solve each linear absolute value equation. Show your work. a. x (x 1 7) 5 3 (x 1 7) 5 3 x x x x x 5 4 x 5 10 b. x (x 9) 5 1 (x 9) 5 1 x x x x x 5 1 x 5 3 c. 3x No solution. Linear absolute value can never be a negative number. d. x x x x 5 3 x 5 3 x Cho, Steve, Artie, and Donald each solved the equation x Artie x 4 = 5 (x) 4 = 5 (x) 4 = 5 (x) = 9 x = 9 x = 9 Cho x 4 = 5 (x) 4 = 5 [(x) 4] = 5 x 4 = 5 x + 4 = 5 x = 9 x = 1 x = 1 Donald x 4 = 5 x = 9 (x) = 9 (x) = 9 x = 9 Steve Before you start solving each equation, think about the number of solutions each equation may have. You may be able to save yourself some work and time!. x 4 = 5 (x) 4 = +5 (x) 4 = 5 x = 9 x 4 = 5 x = 1 x = 1 18 Chapter Graphs, Equations, and Inequalities

89 a. Explain how Cho and Steve incorrectly rewrote the absolute value equation as two separate equations. Cho incorrectly applied the negative symbol to the (4) in the second equation. Steve incorrectly applied a second negative symbol to the (5). b. Explain the difference in the strategies that Artie and Donald used. Which strategy do you prefer? Why? Artie applied the positive and negative signs to the expression inside the linear absolute value symbols and then solved each equation. Donald isolated the linear absolute value expression before applying the positive and negative symbols to the isolated expression. 8. Solve each linear absolute value equation. a. x x x 5 16 x 5 16 (x) 5 16 x 5 16 Consider isolating the absolute value part of the equation before you rewrite as two equations. b. 3 5 x (x 8) (x 8) (x 8) 17 5 (x 8) 5 5 x 17 5 x x c. 3 x x x 5 4 (x ) 5 4 (x ) 5 4 x 5 4 x 5 6 x 5 d x x x x x (x 1 6) 9 5 (x 1 6) 9 5 x x 15 5 x.5 Absolute Value Equations and Inequalities 1.5 Absolute Value Equations and Inequalities 19

90 Problem 3 The scenario from Problem is used again. This time students will write a linear absolute value inequality to describe all of the acceptable weights that the baseball could be and still fit within the specifications. Students then use the inequality to answer several questions and graph the solutions on a number line. They also write and graph an inequality to represent all of the baseballs that do not satisfy the specifications. Another scenario describes acceptable diameters of baseballs and students complete a table of values, write the linear absolute value expression and use the graphing calculator to draw a graph. The graph is used to estimate all of the baseballs that fit within specifications and then verified using algebra. PROBLEM 3 too Big? too Small? Just right. In Too Heavy? Too Light? You re Out! you determined the linear absolute value equation to identify the most and least a baseball could weigh and still be within the specifications. The manufacturer wants to determine all of the acceptable weights that the baseball could be and still fit within the specifications. You can write a linear absolute value inequality to represent this problem situation. 1. Write a linear absolute value inequality to represent all baseball weights that are within the specifications. w # Determine if each baseball has an acceptable weight. Explain your reasoning. a. A manufactured baseball weighs 147 grams. w # # # # 3.95 Yes. This baseball is an acceptable weight because the difference is only grams and it can have a difference up to 3.95 grams. b. A manufactured baseball weighs grams. w # # # No. This baseball is not an acceptable weight because the difference is 4.45 grams which is greater than the specified difference of 3.95 grams. Grouping Ask a student to read the information. Discuss as a class. Have students complete Questions 1 through 4 with a partner. Then share the responses as a class. Guiding Questions for Discuss Phase, Questions 1 and To fit the specifications, the baseball must be at least what weight? To fit the specifications, the baseball must be at most what weight? c. A manufactured baseball weighs grams. w # # # Yes. This baseball is an acceptable weight because the difference is 3.95 grams which is exactly the specified difference. 130 Chapter Graphs, Equations, and Inequalities

91 Guiding Questions for Share Phase Questions 3 and 4 What is the absolute value of equal to? What is the absolute value of equal to? What is the absolute value of equal to? Are the circles on the graph of the acceptable weights open or closed? Why? Does your graph describe all of the acceptable baseball weights? How do you know? Are the circles on the graph of the non-acceptable weights open or closed? Why? Does your graph describe all of the non-acceptable baseball weights? How do you know? What is the relationship between the graph of the acceptable weights and the graph of the non-acceptable weights? d. A manufactured baseball weighs grams. w # # # Yes. This baseball is an acceptable weight because the difference is 3.95 grams, which is exactly the specified difference. 3. Complete the inequality to describe all the acceptable weights, where w is the baseball s weight. Then use the number line to graph this inequality # w # Raymond has the job of disposing of all baseballs that are not within the acceptable weight limits. a. Write an inequality to represent the weights of baseballs that Raymond can dispose of. w, or w b. Graph the inequality on the number line Absolute Value Equations and Inequalities 13.5 Absolute Value Equations and Inequalities 131

92 Grouping Have students complete Questions 5 through 6 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Question 5 What is the target diameter? How is it used to solve the problem? How is this scenario similar to the last scenario? How is this scenario different from the last scenario? What is the unit of measure for the diameter of the baseball? What unit of measure is used to represent the difference between the diameter of the ball and the target diameter of the ball? How is the target diameter used to determine the difference? When the diameter was 6.54 centimeters, how was the target diameter used to determine the difference? In Little League Baseball, the diameter of the ball is slightly smaller than that of a professional baseball. 5. The same manufacturer also makes Little League baseballs. For these baseballs, the manufacturer sets the target diameter to be 7.47 centimeters. The specified diameter allows for a difference of 1.7 centimeters. a. Denise measures the diameter of the Little League baseballs as they are being made. Complete the table to determine each difference. Then write the linear absolute value expression used to determine the diameter differences. Independent Quantity Diameter of the Little League Baseballs Dependent Quantity Target and Actual Diameter Difference Units centimeters centimeters d d Chapter Graphs, Equations, and Inequalities

93 Guiding Questions for Share Phase, Questions 5 through 7 What is the shape of the graph? Why do you think it is this shape? Where does the graph of y intersect the original graph? Does the graph of y intersect the original graph at more than one point? Why does the graph of y intersect the original graph at two different points? What inequality best represents this situation? How can the box and oval method be used to verify your solution? Be sure to label your axes. b. Graph the linear absolute value function, f(d ), on a graphing calculator. Sketch the graph on the coordinate plane. Diameter Difference (cm) y y y x Diameter of Little League Baseballs (cm) 6. Determine the diameters of all Little League baseballs that fit within the specifications. a. Use your graph to estimate the diameters of all the Little League baseballs that fit within the specifications. Explain how you determined your answer. The graph of y crosses the original graph at two points. The first is between 6 and 6.5 so I would estimate the smallest diameter of a baseball that meets specifications would be about 6.5 centimeters. It also crosses between 8.5 and 9 so I would estimate the largest diameter of a baseball that meets the specifications would be about 8.74 centimeters. I chose 8.74 instead of 8.75 because I already know 8.75 is too large from the table. b. Algebraically determine the diameters of all the baseballs that fit within the specification. Write your answer as an inequality. d 7.47 # # d 7.47 # # d # 8.74 The baseballs can have a diameter between 6. and 8.74 centimeters and fit within the specifications..5 Absolute Value Equations and Inequalities 13.5 Absolute Value Equations and Inequalities 133

94 7. The manufacturer knows that the closer the diameter of the baseball is to the target, the more likely it is to be sold. The manufacturer decides to only keep the baseballs that are less than 0.75 centimeter from the target diameter. a. Algebraically determine which baseballs will not fall within the new specified limits and will not be kept. Write your answer as an inequality. d 7.47 $ 0.75 d 7.47 # 0.75 d 7.47 $ 0.75 or d # 6.7 d $ 8. b. How can you use your graph to determine if you are correct? My graph can help me determine if I am correct by using the box and oval method. I can see that the numbers that are outside of the specified range are all numbers less than about 6.5, and all numbers greater than about 8. My solutions fit within that range. Talk the Talk Linear absolute value inequalities are solved and graphed on number lines by rewriting them as equivalent compound inequalities. talk the talk Absolute value inequalities can take four different forms as shown in the table. To solve a linear absolute value inequality, you must first write it as an equivalent compound inequality. Notice that the equivalent compound inequalities do not contain absolute values. Absolute Value Inequality ax 1 b, c Equivalent Compound Inequality c, ax 1 b, c Grouping Ask a student to read the definition. Discuss the table as a class. Have students complete Question 1 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Question 1 How do you rewrite the linear absolute value inequality as an equivalent compound inequality? Is it rewritten in compact form? Are the circles on the graph open or closed? Why? ax 1 b # c ax 1 b. c ax 1 b $ c c # ax 1 b # c ax 1 b, c or ax 1 b. c ax 1 b # c or ax 1 b $ c 1. Solve the linear absolute value inequality by rewriting it as an equivalent compound inequality. Then graph your solution on the number line. a. x 1 3, , x 1 3, 4 7, x, 1 As a final step, don t forget to check your solution. How would you explain to someone how to rewrite the linear absolute value inequality? How do you know when and where to place the negative sign in the inequality? How do you know which inequality sign you need to use? 134 Chapter Graphs, Equations, and Inequalities

95 b. 6 # x # x 4 6 $ x 4 10 # x $ x 5 # x or 1 $ x c. 5x 1 8 1, x 1 8, 3 3, 5x 1 8, 3 3 8, 5x, , 5x, 15 3, x, 31 5 d. x The solution is all numbers. The absolute value of any number is always greater than a negative number. e. x 1 5, There is no solution. The absolute value can never be less than a negative number. Be prepared to share your solutions and methods..5 Absolute Value Equations and Inequalities 13.5 Absolute Value Equations and Inequalities 135

96 Check for Students Understanding Match each inequality with the correct solution. 1. (B). (D) 4x 1 1, 0 A. x, 55, x B. x, 3. (A) 3 x C. x. 4. (C) 8x, 16 D. x. 136 Chapter Graphs, Equations, and Inequalities

97 Choose Wisely! Understanding Non-Linear Graphs and Inequalities.6 Learning Goals In this lesson, you will: Identify the appropriate function to represent a problem situation. Determine solutions to linear functions using intersection points. Determine solutions to non-linear functions using intersection points. Describe advantages and disadvantages of using technology different methods to solve functions with and without technology. Essential Ideas Scenarios are represented by using a linear function, an exponential function, a linear absolute value function, and a quadratic function. Points of intersection are used to graphically determine solutions to non-linear functions. The graphing calculator is used to graph functions and determine specific features of the function such as points of intersection, intercepts and maximum value. Mathematics Common Core Standards N-Q Quantities Reason quantitatively and use units to solve problems. 1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.. Define appropriate quantities for the purpose of descriptive modeling. A-CED Creating Equations Create equations that describe numbers or relationships. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. A-REI Reasoning with Equations and Inequalities Represent and solve equations and inequalities graphically 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 137A

98 F-IF Interpreting Functions Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F-LE Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems 1. Distinguish between situations that can be modeled with linear functions and with exponential functions. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Overview Scenarios described by a linear function, an exponential function, a linear absolute value function, and a quadratic function are given. Students will choose the appropriate function to represent the situation, complete a table of values, graph the function, and solve for values of the functions. The intersection feature of the graphing calculator is used to determine an exact answer. Students calculate answers with and without the use of technology. Finally, they will complete tables that describe each solution method used in this lesson from a numeric, graphic, and algebraic perspective. They also highlight the advantages and limitations with and without the use of technology for each method. 137B Chapter Graphs, Equations, and Inequalities

99 Warm Up A complete graph shows the overall behavior of the function. Use a graphing calculator in each question. 1. Enter the quadratic function f(x) 5 x 17x 1 60 on a graphing calculator.. Press Zoom and select option 6 (ZStandard) 3. The graph of the function is not a complete graph. Explain your reasoning. The graph is not complete because you cannot view the minimum of the parabola, you cannot view one of the x-intercepts, you cannot view the y-intercept and you cannot view the right side of the parabola. 4. Describe what needs to be adjusted to view a complete graph of this quadratic function. More negative y-values are needed to view the minimum of the graph of the quadratic function, more positive y-values are needed to view the y-intercept and more positive x-values are needed to view the second x-intercept and right side of the parabola. This can be done by pressing Window and adjusting the Ymin to a value less than 1, adjusting the Ymax to a value greater than 60 and adjusting the Xmax to a value greater than To view a complete graph, describe a reasonable viewing window. WINDOW Xmin 5 10 Ymin 5 16 Xscl 5 1 Ymin 5 13 Ymax 5 61 Yscl 5 1 Answers may vary..6 Understanding Non-Linear Graphs and Inequalities 137C

100 137D Chapter Graphs, Equations, and Inequalities

101 Choose Wisely! Understanding non-linear graphs and inequalities learning goals In this lesson, you will: Identify the appropriate function to represent a problem situation. Determine solutions to linear functions using intersection points. Determine solutions to non-linear functions using intersection points. Describe advantages and disadvantages of using technology different methods to solve functions with and without technology..1.6 We make decisions constantly: what time to wake up, what clothes to wear to school, whether or not to eat a big or small breakfast. And those decisions all happen a few hours after you wake up! So how do we decide what we do? There are actually a few different techniques for making decisions. One technique, which you have most likely heard about from a teacher, is weighing the pros and cons of your options then choosing the one that will result in the best outcome. Another technique is called satisficing which means just using the first acceptable option, which probably isn t the best technique. Have you ever flipped a coin to make a decision? That is called flipism. Finally, some people may follow a person they deem an expert while others do the most opposite action recommended by experts. While the technique you use isn t really important for some decisions (flipping a coin to decide whether or not to watch a TV show), there are plenty of decisions where there is a definite better choice (do you really want to flip a coin to decide whether to wear your pajamas to school?). The best advice for making decisions is to know your goal, gather all the information you can, determine pros and cons of each alternative decision, and make the decision. What technique do you use when making decisions? Do you think some people are better decision makers than others? What makes them so?.6 Understanding Non-Linear Graphs and Inequalities 137

102 Problem 1 A scenario that describes a linear function is given. Students will choose the appropriate function, complete a table of values, and use the table to create a graph of the situation. The graph is then used to estimate a value on the function, and the intersection feature on a graphing calculator is used to determine exact values of the intersection of a horizontal line and the function. Grouping Have students complete Questions 1 through 5 with a partner. Then share the responses as a class. PROBLEM 1 grill Em Up! Your family is holding their annual cookout and you are in charge of buying food. On the menu are hamburgers and hot dogs. You have a budget determining how much you can spend. You have already purchased 3 packs of hot dogs at $.9 a pack. You also need to buy the ground meat for the hamburgers. Ground meat sells for $.99 per pound, but you are unsure of how many pounds to buy. You must determine the total cost of your shopping trip to know if you stayed within your budget. This problem situation is represented by one of the following functions: f(p) 5.99p f(p) 5.9 p p f(p) 5.99p f(p) 5 3 p 1.9p Choose a function to represent this problem situation. Explain your reasoning. The linear function f(p) 5.99p represents this problem situation.. Complete the table to represent the total amount paid as a function of the amount of ground meat purchased. Don t forget to determine the units of measure. Guiding Questions for Share Phase, Questions 1 and What function family represents this problem situation? Is the function written in the form of f(x) 5 ax 1 b? What is a? What is b? Is the slope negative or positive? What in the scenario represents the independent quantity? What in the scenario represents the dependent quantity? What unit of measure is used to express the amount of ground meat? What unit of measure is used to represent the total amount? Quantity Independent Quantity Amount of Ground Meat Dependent Quantity Total Cost Units pounds dollars Expression p.99p Chapter Graphs, Equations, and Inequalities

103 Guiding Questions for Share Phase, Questions 3 through 5 When the total cost was $13.60, how did you determine the amount of ground meat? How do you determine the y-intercept using the expression representing the total cost? What is the y-intercept on the graph of the function? What does the y-intercept represent with respect to the problem situation? Was a vertical or horizontal line used to represent the total bill of $13.45? What are the coordinates of the point determined where the two lines appear to intersect? What is the significance of this point of intersection with respect to this problem situation? How does the intersection feature on a graphing calculator help to determine the exact coordinates of the point of intersection? How did your estimate compare to the exact answer? How will you determine if you have enough money? How will you determine if you have any money left over? 3. Use the data from the table to create a graph of the problem situation on the coordinate plane. Total Cost (dollars) y Amount of Ground Meat (pounds) 4. Consider a total bill of $ a. Estimate the amount of ground beef purchased. The graph of y crosses the original graph at about, so I predict that pounds of ground meat were purchased. b. Determine the exact amount of ground meat purchased. Using the intersection function on my graphing calculator, I determined the exact amount to be. pounds of ground meat. 5. Based on the number of people coming to the cookout, you decide to buy 6 pounds of ground meat for the hamburgers. a. If your budget for the food is $5.00, do you have enough money? Why or why not? The total cost of the bill will be $4.81, so $5.00 is enough money. b. If you have enough money, how much money do you have left over? If you do not have enough money, how much more will you need? The bill will be $4.81, so I will have $0.19 left over. y x.6 Understanding Non-Linear Graphs and Inequalities 13.6 Understanding Non-Linear Graphs and Inequalities 139

104 Problem A scenario that describes an exponential function is given. Students will choose the appropriate function, complete a table of values, and use the table to create a graph of the situation. The graph is then used to estimate a value on the function, and the intersection feature on a graphing calculator is used to determine exact values of the intersection of a horizontal line and the function. Grouping Have students complete Questions 1 through 5 with a partner. Then share the responses as a class. PROBLEM ground Breaking Costs A construction company bought a new bulldozer for $15,000. The company estimates that its heavy equipment loses one-fifth of its value each year. This problem situation is represented by one of the following functions: f(t) 5 15,000t 1 5 f(t) 5 15,000 ( 4 5 ) t f(t) t 15,000 f(t) 5 t 1 15,000t Choose a function to represent this problem situation. Explain your reasoning. The function f(t) 5 15,000 ( 4 5 ) t represents this problem situation. I know this is the correct function because it represents an exponential function and the problem situation is an exponential situation.. Complete the table to represent the cost of the bulldozer as a function of the number of years it is owned. Quantity Independent Quantity Time Owned Dependent Quantity Total Value of the Bulldozer Guiding Questions for Share Phase, Questions 1 and If the heavy equipment loses one-fifth of its value each year, how much of a fractional value does the heavy equipment retain each year? What function family represents this problem situation? Is the function written in the form of f(x) 5 a(b) x? What is a? What is b? What in the scenario represents the independent quantity? What in the scenario represents the dependent quantity? Units years dollars Expression t 15,000 ( 4 5 ) t 0 15, , , , , , What unit of measure is used to express the time the bulldozer was owned? What unit of measure is used to represent the value of the bulldozer? When the number of years the bulldozer was owned was 0, how did you determine the value of the bulldozer? When the number of years the bulldozer was owned was.5, how did you determine the value of the bulldozer? What is the y-intercept on the graph of the function? 140 Chapter Graphs, Equations, and Inequalities

105 Guiding Questions for Share Phase Questions 3 through 5 What does the y-intercept represent with respect to the problem situation? Was a vertical or horizontal line used to represent the value of $5,000? Where did you draw the oval on the graph of the function? What are the coordinates of the point determined where the graphs of the two functions appear to intersect? What is the significance of this point of intersection with respect to this problem situation? How does the intersection feature on a graphing calculator help to determine the exact coordinates of the point of intersection? How did your estimate compare to the exact answer? What numbers were used when writing your answer as an inequality? What signs were used when writing your answer as an inequality? How did you determine when the bulldozer will be worth $0? 3. Use the data and the function to graph the problem situation on the coordinate plane shown. 4. The owner wants to sell the bulldozer and make at least $5,000 in the sale. a. Estimate the amount of time the owner has to achieve this goal. The graph of y 5 5,000 crosses the original graph between 6.5 and 7.5, so I estimate the company can sell the bulldozer any time before they have owned it for 7 years. t, 7 Total Value of the Bulldozer (dollars) b. Determine the exact amount of time the owner has to achieve this goal. Write your answer as an inequality. Using the intersection function on my graphing calculator, I determined that the bulldozer will be worth $5,000 when it has been owned for 7. years. The owner can sell it any time before that and still earn at least $5, # t # 7. 11, ,000 87,500 75,000 6,500 50,000 37,500 5,000 1,500 0 y Time Owned (years) 5. When will the bulldozer be worth $0? The bulldozer will never be worth $0. If I use my table function, I can see that the values become smaller and smaller decimals, but the values never become 0..6 Understanding Non-Linear Graphs and Inequalities 14 x.6 Understanding Non-Linear Graphs and Inequalities 141

106 Problem 3 A scenario that describes a linear absolute value function is given. Students will choose the appropriate function, complete a table of values, and use the table to create a graph of the situation. The graph is then used to estimate a value on the function, and the intersection feature on a graphing calculator is used to determine exact values of the intersection of a horizontal line and the function. Grouping Have students complete Questions 1 through 5 with a partner. Then share the responses as a class. PROBLEM 3 Stick the landing! In gymnastics, it is important to have a mat below the equipment to absorb the impact when landing or falling. The thickness of the mats used in the rings, parallel bars, and vault events must be between 7.5 and 8.5 inches thick, with a target thickness of inches. This problem situation is represented by one of the following functions: f(t) t f(t) t f(t) 5 t f(t) t 1 7.5t Choose a function to represent this problem situation. Explain your reasoning. The linear absolute value function f(t) 5 t represents this problem situation.. Complete the table to represent the mat thickness in terms of the target thickness of the mat. Quantity Independent Variable Mat Thickness Dependent Variable Difference From Target Thickness Guiding Questions for Share Phase, Questions 1 and What function family represents this problem situation? Is the function written in the form of f(x) 5 x a? What is a? What in the scenario represents the independent quantity? What in the scenario represents the dependent quantity? What is the target thickness of the mat? How is it used to solve the problem? What unit of measure is used to express the thickness of the mats? Units inches inches Expression t t What unit of measure is used to represent the difference between the actual thickness of the mat and the target thickness of the mat? Is there an absolute value sign in the expression representing the values in the table? When the thickness of the mat was 5.5 inches, how did you determine the difference between the actual thickness of the mat and the target thickness of the mat? 14 Chapter Graphs, Equations, and Inequalities

107 Guiding Questions for Share Phase Questions 3 through 5 When the difference between the actual thickness of the mat and the target thickness of the mat was inches, how did you determine thickness of the mat? How would you describe the shape of the graph of the function? Did you draw more than one oval on the graph of the function? Why? Where did you draw the ovals on the graph of the function? What numbers were used to write the absolute value inequality that represents this situation? What sign was used to write the absolute value inequality that represents this situation? What two equations were written from the absolute vale inequality to determine the thickest and thinnest mats that would be acceptable? What were the two equations used to determine the possible thicknesses of the Gymnastics Club s mat? 3. Use the data and the function to graph the problem situation on the coordinate plane shown. Difference from Target Thickness (inches) y Mat Thickness (inches) 4. The Olympics Committee announces that they will only use mats with a thickness of inches and an acceptable difference of inch. a. Write the absolute value inequality that represents this situation. t # b. Determine the thickest and thinnest mats that will be acceptable for competition. Write your solution as a compound inequality. t t t t # t # The All-Star Gymnastics Club has a practice mat with a thickness that is 1.65 inches off the Olympic recommendations. What are the possible thicknesses of the Gymnastics Club s practice mat? t t t or t The thickness of the mat is either 6.5 inches or 9.5 inches..6 Understanding Non-Linear Graphs and Inequalities 14 x.6 Understanding Non-Linear Graphs and Inequalities 143

108 Problem 4 A scenario that describes a quadratic function is given. Students will choose the appropriate function, complete a table of values, and use the table to create a graph of the situation. The graph is then used to estimate two values on the function, and the intersection feature on a graphing calculator is used to determine exact values of the intersection of a horizontal line and the function. PROBLEM 4 Fore! In 1971, astronaut Alan Shepard hit a golf ball on the moon. He hit the ball at an angle of 45 with a speed of 100 feet per second. The acceleration of the ball due to the gravity on the moon is 5.3 feet per second squared. Then the ball landed. This problem situation is represented by one of the following functions: f(d) 5 5.3d f(d) d f(d) 5 5.3d f(d) 5 10,000 d 1 d 1. Choose a function to represent this problem situation. 5.3 The problem situation is represented by the quadratic function f(d) 5 10,000 d 1 d. Note Students may have trouble choosing the correct function to represent this problem situation. Have a discussion and point out that the correct function should describe the height of the golf ball in terms of the distance the golf ball has traveled. They may not recognize this is a quadratic situation. Depending on their prior experiences, vertical motion may be a relatively unfamiliar topic. A mathematical model for a vertical motion problem on earth using height, time, and velocity is written as h 5 16t 1 v o t 1 h 0, where t is the time in seconds, v 0 is the initial velocity, and h 0 is the initial height of the object. Grouping Have students complete Questions 1 through 6 with a partner. Then share the responses as a class.. Complete the table to represent the height of the golf ball in terms of the distance it was hit. Quantity Independent Quantity Horizontal Distance of the Golf Ball Dependent Quantity Height of the Golf Ball Units feet feet 5.3 Expression d 10,000 d 1 d Guiding Questions for Discuss Phase, Questions 1 and What function family represents this problem situation? Is the function written in the form of f(x) 5 ax 1 bx 1 c? What is a? What in the scenario represents the independent quantity? What in the scenario represents the dependent quantity? What unit of measure is used to express the distance the golf ball travels? What unit of measure is used to represent the height of the golf ball? 144 Chapter Graphs, Equations, and Inequalities

109 Guiding Questions for Share Phase, Questions 3 through 5 When the golf ball traveled 405 feet, how did you determine the height of the golf ball? What is the y-intercept on the graph of the function? What does the y-intercept represent with respect to this problem situation? Was a vertical or horizontal line used to represent the distance of 363 feet? What are the coordinates of the points determined where the graph of the horizontal line appears to intersect the graph of the quadratic function? What is the significance of these points of intersection with respect to this problem situation? How does the intersection feature on a graphing calculator help to determine the exact coordinates of the points of intersection? Where on the graph of the quadratic function is the maximum height? What feature on a graphing calculator is used to determine the maximum height of the golf ball? What do the x-intercepts represent with respect to this problem situation? talk the talk Talk the Talk 3. Use the data and the function to graph the problem situation on the coordinate plane shown. Height of the Golf Ball (feet) y y x Horizontal Distance of the Golf Ball (feet) 4. The Saturn V rocket that launched Alan Shepard into space was 363 feet tall. At what horizontal distance was the golf ball higher than the rocket was tall? Using the intersection function on my calculator, I determined that the golf ball was higher than 363 feet at feet and at feet, so anywhere between those distances the ball was higher than the rocket was tall. 5. At what horizontal distance did the golf ball reach its maximum height? What was the greatest height the ball reached? The ball reached its greatest height of feet when it was at a horizontal distance of feet. 6. How far did the golf ball travel before it landed back on the moon? The golf ball traveled a total distance of feet before landing back on the moon. In this chapter you used three different methods to determine values of various functions. You completed numeric tables of values, determined values from graphs, and solved equations algebraically. In addition, you used each of these methods by hand and with a graphing calculator. Think about each of the various methods for problem solving and complete the tables on the following pages. Pay attention to the unknown when describing each strategy. Don t forget you have worked with linear functions, exponential functions, and quadratic functions. Keep all three in mind when completing the tables..6 Understanding Non-Linear Graphs and Inequalities 14 Students will describe all of the methods they used in this lesson to determine values of various functions with and without the use of technology. This includes completing numeric tables of values, determining values from graphs, and solving equations algebraically..6 Understanding Non-Linear Graphs and Inequalities 145

110 Grouping Have students complete the tables with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Talk the Talk If you are given the independent quantity in a table of values and the scenario, can you compute the values of the dependent quantity numerically? How is it done without the use of technology? How is it done with the use of technology? If you are given the dependent quantity in a table of values and the scenario, can you compute the values of the independent quantity numerically? How is it done without the use of technology? How is it done with the use of technology? Given an Independent Quantity (input value) Given a Dependent Quantity (output value) Numerically Without Technology With Technology Description of the method: Description of the method: Using the independent quantities already Using the independent quantities given in given in the table and the information from the table, input the values into the function the problem situation, determine the and solve for the output or create a table output values and then determine the using the TBLSET function to set the function. Or, if the function is given, use it intervals and starting value. to complete the table. Advantages: Provides an exact solution for given values. Provides a range of solutions depending on the intervals of the table. Disadvantages/Limitation: May be unable to solve or determine more difficult functions. You are limited by the intervals of the input values. May need to estimate for values that fall between intervals. Description of the method: Using the dependent quantities already given in the table and the information from the problem situation, use the function to determine the input values. Advantages: Provides an exact solution for given values. Provides a range of solution dependent on the intervals of the table. Advantages: Provides an exact solution for given intervals. Able to solve more mathematically difficult functions. Provides wide range of inputs if using the TBLSET function. Disadvantages: You are limited by the intervals of the TBLSET. May need to estimate for values that fall between intervals. Description of the method: Using the dependent quantities given in the table, solve for the independent values. Advantages: Provides an exact solution for independent values. May be able to solve more difficult functions with technology. Disadvantages/Limitations: May be unable to solve more difficult functions due to current skill levels. Table is limited to the intervals and values provided. May need to estimate for values that fall between intervals. Disadvantages/Limitations: Solving for the independent values can be complicated and may lead to errors for more difficult functions. 146 Chapter Graphs, Equations, and Inequalities

111 Guiding Questions for Share Phase, Talk the Talk If you are given the independent quantity in a table of values and the scenario, can you compute the values of the dependent quantity graphically? How is it done without the use of technology? How is it done with the use of technology? If you are given the dependent quantity in a table of values and the scenario, can you compute the values of the independent quantity graphically? How is it done without the use of technology? How is it done with the use of technology? Given an Independent Quantity (input value) Given a Dependent Quantity (output value) Graphically Without Technology With Technology Description of the method: Description of the method: Graph the function. Locate the input Graph the function. Use the value feature value on the x-axis and determine the to enter the input value. The calculator corresponding y-value which is the produces the output value. output value. Advantages: Provides a visual representation of the function. Can provide a wide range of values depending upon the intervals. Disadvantages/Limitations: Often have to estimate values on a graph. If points do not fall directly on intervals, you are unable to determine an exact solution. Also, the number of values is limited by the bounds of the graph. Description of the method: Graph the function and the dependent quantity. Locate the intersection point. The x-coordinate of the intersection point represents the input value. Advantages: Provides a visual representation of the function. Can determine exact output value for any input value. Can provide a wide range of values depending upon the intervals. Disadvantages/Limitations: May need to adjust the bounds of the window to view necessary values on the graph. Description of the method: Graph the two functions. Use the intersect feature to determine the exact intersection point. Advantages: Provides a visual representation of the function. Can determine at least an estimate of the value. Advantages: Provides a visual representation of the function. Provides an exact solution for any intersection point. Disadvantages/Limitations: Often have to estimate values on a graph. Unless the intersection point falls directly on an interval, you are unable to determine an exact solution. Disadvantages/Limitations: May need to adjust the bounds of the window to view all data points..6 Understanding Non-Linear Graphs and Inequalities 14.6 Understanding Non-Linear Graphs and Inequalities 147

112 Guiding Questions for Share Phase, Talk the Talk If you are given the independent quantity in a table of values and the scenario, can you compute the values of the dependent quantity algebraically? How is it done without the use of technology? How is it done with the use of technology? If you are given the dependent quantity in a table of values and the scenario, can you compute the values of the independent quantity algebraically? How is it done without the use of technology? How is it done with the use of technology? Given an Independent Quantity (input value) Given a Dependent Quantity (output value) Algebraically Without Technology With Technology Description of method: Description of method: Using the given function, replace the Replace the variables in the function with variables in the function with the input the given input value. Calculate the value. Simplify the function. solution. Advantages: Provides an exact solution for every input. Disadvantages/Limitations: May be unable to solve more difficult functions due to lack of knowledge. Description of method: Replace the function notation with the given dependent quantity and solve for the independent quantity. Advantages: Provides an exact solution for every input. Allows solving for more difficult functions. Disadvantages/Limitations: May still be unable to solve more difficult functions. Description of method: Rearrange the equation to solve for the variable. Then replace the function with its value and solve. Advantages: Provides an exact solution for every input. Advantages: Provides an exact solution for every input. Disadvantages/Limitations: May be unable to solve non-linear functions at this time due to lack of knowledge. Disadvantages/Limitations: Rearranging the equation can be complicated and may lead to errors. Be prepared to share your solutions and methods. 148 Chapter Graphs, Equations, and Inequalities

113 Check for Students Understanding 1. Use a graphing calculator to answer each question for the quadratic function g(x) 5 x 1 14x 5. a. To view a complete graph of the function, what is a reasonable viewing window? WINDOW Xmin 5 0 Ymin 5 10 Xscl 5 1 Ymin 5 80 Ymax 5 10 Yscl 5 10 Answers may vary. b. Sketch a graph of the function on the coordinate plane shown. y x c. Determine the y-intercept. The y-intercept is approximately (0, 4.99). d. Determine the x-intercepts. The x-intercepts are approximately (15.60, 0) and (1.60, 0). e. Determine the lowest minimum point. The minimum point is approximately (6.999, 74)..6 Understanding Non-Linear Graphs and Inequalities 148A

114 148B Chapter Graphs, Equations, and Inequalities

115 Chapter Summary KEy terms first differences (.1) solution (.1) intersection point (.1) solve an inequality (.3) compound inequality (.4) solution of a compound inequality (.4) conjunction (.4) disjunction (.4) opposites (.5) absolute value (.5) linear absolute value equation (.5) linear absolute value inequality (.5) equivalent compound inequality (.5).1 identifying Dependent and independent Quantities and Writing an Expression The dependent quantity is dependent on how the independent quantity changes. The independent quantities are the input values of an expression and the dependent quantities are the output values. Example The table of values identifies the independent and dependent quantities and their units for the problem situation. An expression for the dependent quantity is written based on the independent quantity variable. Caroline earns $5 a week babysitting after school. She deposits half of this amount in her savings account every Saturday. Independent Quantity Dependent Quantity Quantity Time Money Saved Units weeks dollars Expression w 1.5w 149

116 .1 Determining the Unit rate of Change One way to determine the unit rate of change is to calculate first differences. First differences are calculated by taking the difference between successive points. Another way to determine the unit rate of change is to calculate the rate of change between any two ordered pairs and then write each rate with a denominator of 1. With two ordered pairs, the rate of change is the difference between the output values over the difference between the input values. Example Using first differences, the rate of change is Time (weeks) Money Saved (dollars) First Differences Using two ordered pairs, the rate of change is (, 5.00) and (5, 6.50) Determining the Solution to a linear Equation Using Function notation To write a linear equation in function notation, f(x) 5 ax 1 b, identify the dependent (output value) and independent (input value) quantities and the rate of change in a problem situation. Determine a solution to the equation by substituting a value for the independent quantity in the equation. Example Caroline earns $5 a week babysitting after school. She deposits half of this amount in her savings account every Saturday. s(w) 5 1.5w s(14) 5 1.5(14) Caroline will have $175 saved after 14 weeks. s(14) Chapter Graphs, Equations, and Inequalities