4-6 Graphing and Writing Linear Functions Going Deeper Essential question: How can you represent relationships using linear functions? Math Background Standards for Mathematical Content A linear function of the form f (x) = mx + b can be written if the slope and the y-intercept are known. The slope can be determined as long as the coordinates of two points are known or can be obtained from a graph. Calculate the slope from the coordinates of two points as follows: (x2) - f (x1) rise f____________ m = ____ run = x2 - x1 . You can then find the value of b using m and one of the known points. If the known point is (x1, f (x1)), then b = f (x1) - mx1, or f (x2) - f (x1) b = f (x1) - ____________ x2 - x1 (x1). The function f (x) = mx + b can then be rewritten as f (x) = mx + b = f____________ (x2) - f (x1) f____________ (x2) - f (x1) x2 - x1 (x) + f (x1) - x2 - x1 (x1). A-CED.1.2 Create equations in two … variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.* A-REI.4.11 Explain why the x-coordinates of the points where the graphs of the equations y = f (x) and y = g(x) intersect are the solutions of the equation f (x) = g(x); … Include cases where f (x) and/or g(x) are linear … functions.* F-IF.2.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.* F-IF.2.6 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.* F-IF.3.7 Graph functions expressed symbolically and show key features of the graph ...* F-IF.3.7a Graph linear ... functions and show intercepts ...* F-BF.1.1 Write a function that describes a relationship between two quantities.* F-LE.1.2 Construct linear … functions, … given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table).* IN = . TR OD UCE Ask students to graph the function f(x) = 3x + 4 on graph paper and identify the slope and the y-intercept. Have them identify several points of the graph and verify that each pair of points has the same slope. 1 EXAMPLE Questioning Strategies • Does the line in the graph for part A have positive or negative slope? negative Prerequisites • Does the line in the graph for part A rise or fall as the run moves from left to right? fall Continuous linear functions Slope values of m and b in f(x) = mx + b • Why does the graph in part B have a y-intercept of 1? The pitcher starts out containing 1 cup of juice. continued Chapter 4 215 Lesson 6 © Houghton Mifflin Harcourt Publishing Company TE ACH Name Class Notes 4-6 Date Graphing and Writing Linear Functions Going Deeper Essential question: How can you represent relationships using linear functions? Graphing Lines You can graph the linear function f (x) = mx + b using only the slope m and the y-intercept b. First, locate the point (0, b) on the y-axis. Next, use the rise and run of the slope to locate another point on the line. Draw the line through the two points. When using m to locate a second point, bear in mind that m is a ratio, so many values of 1 , then you could use a rise of __ 1 and a run rise and run are possible. For instance, if m = __ 2 2 of 1, a rise of 1 and run of 2, a rise of -2 and a run of -4, and so on. Your choice of rise and run often depends on the scales used on the coordinate plane’s axes. F-IF.3.7a 1 EXAMPLE Graphing a Line Using the Slope and y -Intercept Graph each function. 2x + 4 f (x) = -__ 3 A 4 . Plot the point that r The y-intercept is corresponds to the y-intercept. r The slope is 3 run is 2 -_ 3 6 . 4 © Houghton Mifflin Harcourt Publishing Company r Use the slope to move from the first point to a second 2 point. Begin by moving down units, because the 3 rise is negative. Then move right units, because the run is positive. Plot a second point. 2 x -4 real 2 4 6 8 -4 numbers. real The range is the set of 0 -2 -2 r Draw the line through the two points. r The domain is the set of B f(x) 8 . If you use -2 as the rise, then the numbers. Amount of liquid (cups) A pitcher with a maximum capacity of 4 cups contains 1 cup of apple juice concentrate. A faucet is turned on filling the pitcher at a rate of 0.25 cup per second. The amount of liquid in the pitcher (in cups) is a function A(t) of the time t (in seconds) that the water is running. r The y-intercept is the initial amount in the pitcher at 1 time 0, or cup. Plot the point that corresponds to the y-intercept. r The slope is the rate of change: 0.25 cup per second, 4 1 or 1 cup in seconds. So, the rise is and 4 . the run is 8 A(t) 6 4 2 t 0 r Use the rise and run to move from the first point to a 1 second point on the line by moving up unit and 4 right units. Plot a second point. 4 8 12 16 Time (seconds) continued 215 Chapter 4 Lesson 6 r Connect the points and extend the line to the maximum 4 value of the function, where A(t) = cups. 1 The range is the set of numbers ≤t≤ 12 . ≤ A(t) ≤ 4 0 r The domain is the set of numbers . 1a. How could you use the slope and y-intercept to graph the function f (x) = 3? Write the rule as f(x) = 0x + 3. Start at the point (0, 3). The slope m = 0, so the rise is 0 when the run is 1. Plot (1, 3) and draw a horizontal line through (0, 3) and (1, 3). 1b. What are the units of the rise in Part B? What are the units of the run? The units of the rise are cups. The units of the run are seconds. 1c. How long does it take to fill the pitcher? Explain. 12 seconds; the input associated with a maximum output of 4 cups is 12 seconds. 1 t + 1? Use units to justify your answer. 1d. Why is the function rule A(t) = __ 4 Sample answer: If the function rule is written as f(x) = mx + b, then m is the rate of change and b is the y-intercept, or the initial value where x = 0. Since the pitcher starts with 1 cup of water at time 0, b = 1 cup. The rate of change is cups 1 0.25 cup per second = _ cup per second, and _____ · seconds + cups = cups. 4 2 second F-LE.1.2 EXAMPLE Writing a Linear Function © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company REFLECT Write the linear function f using the given information. A The graph of the function has a slope of 3 and a y-intercept of -1. A linear function has the form f (x) = mx + b where m is the slope and b is the y-intercept. Substitute So, the function is f (x) = B 3 for m and -1 A different function has the values shown in the table. First calculate the slope using any two ordered pairs from the table. For instance, let ( x1, f ( x1 ) ) = (-1, 5) and ( x2, f ( x2 ) ) = (3, -3). f ( x2 ) - f ( x1 ) m = ____________ x -x 2 1 5 -3 - 3 ( -1 ) = _____________ Chapter 4 Chapter 4 - for b. 3x - 1 . x f(x) -1 5 3 -3 7 -11 Write the slope formula. Substitute values. 216 Lesson 6 216 Lesson 6 1 EXAMPLE 2 continued Questioning Strategies • Why is the graph in part B a segment, not a line? The pitcher starts with 1 cup in it at t = 0 and reaches full capacity at t = 12. The graph is Questioning Strategies • For part B, do you get the same slope no matter which two ordered pairs you choose? yes • For part B, what would be the linear function if you used known points (-1, 5) and (7, -11)? f(x) = -2x + 3 limited by the 4-cup capacity of the pitcher. Technology Students may benefit from using a graphing calculator to see the graph in part A. Ask them 2 X + 4 and then verify that the to graph Y1 = - __ 3 graph matches the one in the example. Also ask them to verify that (0, 4) and (3, 2) are points of the graph either by using the Trace feature or by looking at points in the Table function. Avoid Common Errors When calculating the slope using ordered pairs with negative coordinates, students will sometimes forget the negative sign when subtracting values. For example, for (-1, 5) and (3, -3), x2 - x1 should be 3 - (-1), not 3 - 1. Remind students that they must retain any negative signs in an ordered pair when using the slope formula. () Avoid Common Errors Students may have difficulty graphing a function that has a fractional rate of change as shown in this Example. Remind them that the fraction can be looked at as rise over run, so the rise and run can be used to move from one point to a second point. That is, if the rate of change is expressed as a fraction, use the numerator to move the appropriate number of units up or down (the rise) and use the denominator to move the appropriate number of units right (the run) to plot a second point given the first point. Teaching Strategy Encourage students to use the third point in the table to check that they have the correct function. EXTRA EXAMPLE Write the linear function f using the given information. A. The graph of the function has a slope of -5 and a y-intercept of 1. f (x) = -5x + 1 B. The function has values shown in the table. 2 x f(x) -1 -2 1 6 3 14 © Houghton Mifflin Harcourt Publishing Company EXTRA EXAMPLE Graph each function. A. f (x) = -__1 x - 3 The graph is a line with 2 1 . The domain is all y-intercept -3 and slope -__ real numbers. EXAMPLE f(x) = 4x + 2 B. An oil storage tank with a volume of 6000 gallons starts off full and empties at a rate of 60 gallons per minute. The graph is a line with y-intercept 6000 and slope -60. The maximum value is 6000, and the minimum value is 0. The domain is 0 ≤ x ≤ 100. Chapter 4 217 Lesson 6 Notes -8 = ______ Simplify numerator and denominator. = -2 Simplify fraction. 4 Then find the value of b using the fact that m = f (x) = -2 x + b 5 = -2 -2 and f (-1) = 5. Write the function with the known value of m. ( -1 ) + b Substitute -1 for x and 5 for f(x). 5 = 2 +b Simplify the right side of the equation. 3 =b Solve for b. So, the function is f (x) = -2x + 3 . REFLECT 2a. In Part B, use the ordered pair (7, -11) to check your answer. f(7) = -2(7) + 3 = -14 + 3 = -11 F-LE.1.2 3 EXAMPLE Writing a Linear Function from a Graph The graph shows the increase in pressure (measured in pounds per square inch) as a scuba diver descends from a depth of 10 feet to a depth of 30 feet. What is the pressure on the diver at the water’s surface? Pressure (Ib/in.2) © Houghton Mifflin Harcourt Publishing Company Pressure is the result of the weight of the column of water above the diver as well as the weight of the column of Earth’s atmosphere above the water. Pressure is a linear function of depth. Scuba Dividing 32 P(d) (30, 28.0) 24 (10, 19.1) 16 8 d A 0 Interpret the question. 10 20 30 Depth (feet) Let d represent depth and P represent pressure. At the water’s surface, 0 d= . For this value of d, what meaning does P(d) have in terms of the line that contains the line segment shown on the graph? It is the line’s P-intercept. Find the value of m in P(d) = md + b. Use the fact that P(10) = 19.1 and P(30) = 28.0. B P( d2 ) - P( d1 ) m = _____________ d2 - d1 Write the slope formula. 28.0 - 19.1 = ________________ 30 - 10 Substitute values. continued 217 Chapter 4 8.9 = ______ Simplify numerator and denominator. = 0.445 Write in decimal form. 20 Find the value of b in P(d) = md + b. Use the value of m from Part B as well as the fact that P(10) = 19.1. P(d) = 0.445 d + b 19.1 = 0.445 19.1 = 14.7 ≈ b 4.45 ( Write the function with the known value of m. ) 10 + b +b Substitute 10 for d and 19.1 for P(d). Simplify the right side of the equation. Solve for b. Round to the nearest tenth. So, the pressure at the water’s surface is P(0) = b ≈ 14.7 lb/in.2 REFLECT 3a. Interpret the value of m in the context of the problem. m is the rate of change in pressure with respect to depth. 3b. Write the function P(d) = md + b using the calculated values of m and b. Use the function to find the pressure on the diver at a depth of 20 feet. P(d) = 0.445d + 14.7; P(20) = 23.6 lb/in.2 4 A-REI.4.11 EXAMPLE Writing and Solving a System of Equations © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company C Lesson 6 Mr. Jackson takes a commuter bus from his suburban home to his job in the city. He normally gets on the bus in the town where he lives, but today he is running a little late. He gets to the bus stop 2 minutes after the bus has left. He wants to catch up with the bus by the time it gets to the next stop in a neighboring town 5 miles away. The speed limit on the road connecting the two stops is 40 miles per hour, but Mr. Jackson knows that the bus travels the road at 30 miles per hour. He decides to drive at 40 miles per hour to the next stop. Does he successfully catch the bus there? A Identify the independent and dependent variables, how they are measured, and how you will represent them. time The independent variable is , measured in minutes. Let t represent the time since Mr. Jackson began driving to the next bus stop. The dependent variable is distance , measured in miles. Let d represent the distance traveled. Since you need to track the distances traveled by both Mr. Jackson and the bus, use subscripts: dJ will represent the distance traveled by Mr. Jackson, and dB will represent the distance traveled by the bus. Chapter 4 Chapter 4 218 Lesson 6 218 Lesson 6 3 EXAMPLE Highlighting the Standards Questioning Strategies • Why can you extend the graph so that it intersects the y-axis? The depths of the diver from 0 to 3 EXAMPLE is a real-world problem that includes opportunities for mathematical modeling, reasoning, and computation. It is a good opportunity to address Mathematical Practice Standard 4 (Model with mathematics). Draw students’ attention to the way they interpret the values of m and b. The value of m gives the rate of change in pressure with respect to depth: The pressure on the diver increases 0.445 lb/in.2 for every additional foot of depth. The value of b gives the pressure on the diver at the water’s surface, which is just the pressure due to the weight of the column of Earth’s atmosphere above the water: The pressure at the water’s surface is 14.7 lb/in.2 10 feet are reasonable to include in the domain of the function. • What would be the pressure at a depth of 40 feet? Explain how you found your answer. 32.5 lb/in.2; P(40) = 0.445(40) + 14.7 = 32.5 • Why is the graph a first-quadrant graph only? Only nonnegative domain and range values make sense in this situation. Technology Have students substitute the function in Reflect Question 3b into a graphing calculator and then use the Table feature to find additional points on the graph. Weight (kilograms) EXTRA EXAMPLE The graph shows the weight of a female elephant as she grows from 1 to 3 years. What was the weight of the elephant at birth? 4 w 900 Questioning Strategies • What do the coordinates of the intersection point represent in the context of this situation? The (3, 800.0) 700 500 300 distance at which Mr. Jackson catches up with the bus (4 miles) and the time it took Mr. Jackson to reach that distance (6 minutes). (1, 333.3) 100 • Why are conversions necessary in part B? a 1 2 Age (years) The speeds given are in miles per hour, but the elapsed time is in minutes. Both measures need to use the same unit of time to solve and graph the functions. 3 100 kg • Why is each graph only in the first quadrant? Only nonnegative domain and range values make sense in this situation. EXTRA EXAMPLE You are participating in a 2-mile run for charity. You are at the back of the group of runners and leave the starting line 0.5 minute after the first runners have left. You would like to catch up to the first runners by the time you reach the halfway point of the race at 1 mile. The first runners run at a rate of 7.2 miles per hour, and you run at a rate of 7.5 miles per hour. Do you successfully catch up to the lead runners by the time you reach the halfway point of the event? No; you catch up 12 minutes after the race starts, when both you and the lead runners have run 1.5 miles. Chapter 4 219 Lesson 6 © Houghton Mifflin Harcourt Publishing Company 0 EXAMPLE Notes B Write a distance-traveled function for Mr. Jackson and for the bus. Each function has the form d(t) = rt + d0 where r is the rate of travel and d0 is any initial distance. Although you know the rates of travel, they are given in miles per hour, which is incompatible with the unit of time (minutes). So, you need to convert miles per hour to miles per minute. In the conversions below, express the miles as simplified fractions. 40 miles · ___________ 1 hour = _ 2 mile per minute Mr. Jackson: ________ 60 minutes hour 3 30 miles · ___________ 1 hour = _ 1 mile per minute Bus: ________ 60 minutes hour 2 At the moment Mr. Jackson begins driving to the next bus stop, the bus has traveled for 2 minutes. If you use Mr. Jackson’s position as the starting point, then the initial distance for Mr. Jackson is 0 miles, and the _1 initial distance for the bus is 2 · 2 = 1 mile . So, the distance-traveled functions are: 2 t+ Mr. Jackson: dj(t) = _ 0 3 6 d(t) (6, 4) 4 2 t 0 © Houghton Mifflin Harcourt Publishing Company 2 Determine the value of t for which dJ(t) = dB(t). You can do this by graphing the two functions and seeing where the graphs intersect. Carefully draw the graphs on the coordinate plane below, and label the intersection point. Distance (miles) C 1 t+ Bus: dB(t) = _ 1 2 4 6 8 Time (minutes) 6 The t-coordinate of the point of intersection is , so Mr. Jackson catches up with the bus in 6 minutes . D Check the result against the conditions of the problem, and then answer the problem’s question. 5 The problem states that the next bus stop is miles away, and the 4 graph shows that Mr. Jackson catches up with the bus in So, does Mr. Jackson successfully catch the bus? miles. Yes REFLECT 4a. Explain how you can use algebra rather than a graph to find the time when Mr. Jackson catches up with the bus. Then show that you get the same result. 1 2 1 Solve the equation: _ t+1=_ t: 1 = _ t, so t = 6. 2 3 6 219 Chapter 4 Lesson 6 4b. In terms of the context of the problem, explain why the t-coordinate of the intersection point (and not some other point) determines how long it takes Mr. Jackson to catch up with the bus. The two graphs are distance-traveled graphs, and their intersection point is the only point on the two graphs where the distances are equal. The t-coordinate of PRACTICE Graph each linear function. 1x + 2 2. f (x) = __ 2 1. f (x) = 3x - 4 f(x) f(x) 4 4 2 2 x -4 -2 0 2 x 4 -4 -2 0 -2 -2 -4 -4 2 4 2 4 2 4 4x 4. f (x) = __ 3 3. f (x) = -1 f(x) f(x) 4 4 2 2 x -4 -2 0 2 x 4 -4 -2 -2 -2 -4 -4 1x-3 5. f (x) = __ 4 6. f (x) = -5x + 1 f(x) f(x) 4 4 2 2 x -4 Chapter 4 Chapter 4 -2 0 © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company this point gives the time at which this occurs. 2 x 4 -4 -2 0 -2 -2 -4 -4 220 Lesson 6 220 Lesson 6 CLOSE PR ACTICE Essential Question How can you represent relationships using linear functions? You can represent a linear relationship Where skills are taught graphically or symbolically. If you have the function rule, you can draw the graph using the slope and y-intercept. To write a rule from a graph, find the y-intercept b and use two points on the graph to find the value of m; then substitute those values into f (x) = mx + b. From a verbal description, identify the independent and dependent variables, determine the coordinates of points that fit the situation, and write the function f(x) = mx + b. From a table of values, use two ordered pairs to find the value of m and then use m and an ordered pair to find b; then substitute those values into f (x) = mx + b. Where skills are practiced 1 EXAMPLE EXS. 1–6 2 EXAMPLE EXS. 9–16 3 EXAMPLE EXS. 17–19 4 EXAMPLE EX. 20 Exercises 7–8: Students use the skills taught in the lesson to graph a function and extend these skills to make a prediction about the function. Summarize Have students complete a graphic organizer showing the steps for writing a linear function from two given points. A sample is shown below. Given 2 points: Solve for the slope m. Substitute m and the coordinates of one point into f(x) = mx + b. Solve for the y-intercept b. Substitute m and b into f(x) = mx + b. © Houghton Mifflin Harcourt Publishing Company Chapter 4 221 Lesson 6 Notes Graph each linear function and answer the question. Explain your answer. 300 8. A bamboo plant is 10 centimeters tall at noon and grows at a rate of 5 centimeters every 2 hours. The height (in centimeters) is a function h(t) of the time t it grows. When will the plant be 20 centimeters tall? C(t) h(t) 30 Height (cm) Cost (dollars) 7. A plumber charges $50 for a service call plus $75 per hour. The total of these costs (in dollars) is a function C(t) of the time t (in hours) on the job. For how many hours will the cost be $200? $300? 200 100 20 10 t 0 4 2 t 0 6 4 2 Time (hours) 6 Time (hours) 2 hours; the input value associated with about 4:00 P.M.; the input value an output of $200 is 2 hours; about associated with an output of 20 cm is 3.3 hours; the input associated with an about 4 hours, and 4 hours after noon output of $300 is about 3.3 hours. is 4:00 P.M. Write the linear function f using the given information. 9. The graph of the function has a slope of 4 and a y-intercept of 1. 10. The graph of the function has a slope of 0 and a y-intercept of 6. f(x) = 6 © Houghton Mifflin Harcourt Publishing Company f(x) = 4x + 1 12. The graph of the function has a slope 7 and a y-intercept of 0. of __ 4 11. The graph of the function has a slope 2 and a y-intercept of 5. of -__ 3 7 x f(x) = _ 2 f(x) = -_ x+5 4 3 13. 14. x f(x) x f(x) −3 8 0 −3 0 5 2 0 3 2 4 3 3 x-3 f(x) = _ f(x) = -x + 5 15. 2 16. x f(x) 1 −1 x 5 f(x) −2 2 5 10 −6 3 11 15 −10 4 f(x) = -_ x+2 f(x) = 6x - 7 5 221 Chapter 4 Lesson 6 Write the linear function f using the given information. 17. 18. y 8 y 4 6 2 4 x -4 0 -2 2 4 2 x 0 -2 -4 2 4 6 -2 1 f(x) = -_ x+3 f(x) = 2x - 1 2 19. The graph shows the amount of gas remaining in the gas tank of Mrs. Liu’s car as she drives at a steady speed for 2 hours. How long can she drive before her car runs out of gas? Gas Remaining (gallons) Fuel Consumption a. Interpret the question by describing what aspect of the graph would answer the question. Want to know the t-intercept if the graph were extended to the t-axis. g(t) 16 12 8 4 b. Write a linear function whose graph includes the segment shown. t 0 1 2 3 Time (hours) g(t) = -2t + 10 c. Tell how to use the function to answer the question; then find the answer. Set g(t) equal to 0 and solve for t; she can drive for 5 hours. 20. Jamal and Nathan exercise by running one circuit of a basically circular route that is 5 miles long and takes them past each other’s home. The two boys run in the same direction, and Jamal passes Nathan’s home 12 minutes into his run. Jamal runs at a rate of 7.5 miles per hour while Nathan runs at a rate of 6 miles per hour. If the two boys start running at the same time, when, if ever, will Jamal catch up with Nathan before completing his run? © Houghton Mifflin Harcourt Publishing Company a. Identify the independent and dependent variables, how they are measured, and how you will represent them. Independent variable is time t measured in 10 minutes; dependent variable is distance d measured in miles. b. Write distance-run functions for Jamal and Nathan. 1 1 dj(t) = _ t; dN(t) = __ t + 1.5 8 10 Distance (miles) © Houghton Mifflin Harcourt Publishing Company -2 d(t) 8 (60, 7.5) 6 4 2 c. Graph the functions, find the intersection point, and check the point against the conditions of the problem to answer the question. t 0 20 40 60 80 Time (minutes) Jamal does not catch up with Nathan. Chapter 4 Chapter 4 222 Lesson 6 222 Lesson 6 4 x-4 6. y = _ 3 ADD I T I O N A L P R AC T I C E AND PRO BL E M S O LV I N G Assign these pages to help your students practice and apply important lesson concepts. For additional exercises, see the Student Edition. Answers Additional Practice 1. y = 4x - 3 2. y = -2x 1 x+6 3. y = -_ 3 2 10 + 5 4. 3 = _ 5 5 x-5 7. y = _ 2 ( ) 3=4+b -1 = b ( ) 2 x-1 y= _ 5 5. y = -x + 3 b. slope: 3; number of hours per week; y-int: 22; hours already worked c. 70 hours Problem Solving 1. y = 10x + 300 2. slope: 10, rate of the change of the cost: $10 per student; y-int: 300, the initial fee (the cost for 0 students) 3. $800 Chapter 4 223 4. C 5. J 6. A 7. H Lesson 6 © Houghton Mifflin Harcourt Publishing Company 8. a. y = 3x + 22 Name Class Notes 4-6 Date Additional Practice © Houghton Mifflin Harcourt Publishing Company 223 Chapter 4 Lesson 6 © Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company Problem Solving Chapter 4 Chapter 4 224 Lesson 6 224 Lesson 6

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