graphs, Equations, and inequalities

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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