Essential Question: How can you represent a linear function in a way that reveals its slope and y-intercept?

Transcription

1 Resource Locker L E S S O N 61 Slope-Intercept Form Teas Math Standards The student is epected to: A1B Write linear equations in two variables in various forms, including = m + b, given one point and the slope and given two points Also A1C, A13C Mathematical Processes A11D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, diagrams, graphs, and language as appropriate Language Objective C3, C, I3, I, 3D, G Eplain to a partner how to write a linear function in slope-intercept form ENGAGE Essential Question: How can ou represent a linear function in a wa that reveals its slope and -intercept? You can determine the slope m of the graph of the function and its -intercept b and write the equation = m + b, called the slope-intercept form of the equation PREVIEW: LESSON PERFORMANCE TASK View the Engage section online Discuss the photo and how a gm membership ma require a one-time sign-up fee as well as regular monthl fees Also discuss how a graph of this tpe of data might look Then preview the Lesson Performance Task Name Class Date 6 1 Slope-Intercept Form Essential Question: How can ou represent a linear function in a wa that reveals its slope and -intercept? A1B write linear equations in two variables in various forms, including = m + b Also A1C, A13C Eplore Graphing Lines Given Slope and -intercept Resource Locker Graphs of linear equations can be used to model man real-life situations Given the slope and -intercept, ou can graph the line, and use the graph to answer questions Andrew wants to bu a smart phone that costs $ His parents will pa for the phone, and Andrew will pa them $5 each month until the entire amount is repaid The loan repament represents a linear situation in which the amount that Andrew owes his parents is dependent on the number of paments he has made When =, = $ The rate of change in the amount Andrew owes over time is $5 per month The -intercept of the graph of the equation that represents the situation is The slope is 5 Use the -intercept to plot a point on the graph of the equation The -intercept is, so plot the point (, ) Module 6 7 Lesson 1 Name Class Date 6 1 Slope-Intercept Form Essential Question: a wa that reveals its slope -intercept? A1B write linear equations in two variables in various forms, including =m+b Also A1C, A13C Eplore Graphing Lines Given Slope and -intercept Graphs of linear equations can be used to model man real-life situations Given the slope and -intercept, ou can graph the line, and use the graph to answer questions Andrew wants to bu a smart phone that costs $ His parents will pa for the phone, and Andrew will pa them $5 each month until the entire amount is repaid The loan repament represents a linear situation in which the amount that Andrew owes his parents is dependent on the number of paments he has made When =, = The rate of change in the amount Andrew owes over $5 time is per month The -intercept of the graph of the equation that represents the situation is The slope is Use the -intercept to plot a point on the graph of the equation The -intercept is, so plot the point $ 5 (, ) Module 6 7 Lesson 1 HARDCOVER PAGES 195 Turn to these pages to find this lesson in the hardcover student edition 7 Lesson 61

2 Using the definition of slope, plot a second point Change in 5 Slope = Change in =_ = 5 Start at the point ou plotted Count 5 units down and 1 unit right and plot another point Reflect Draw a line through the points ou plotted Discussion How can ou use the same method to find two more points on that same line? Possible answer: You can begin at the second point, (1, 5), and move 5 units down and 1 unit to the right Then repeat this process beginning at the new point How man months will it take Andrew to pa off his loan? Eplain our answer 1 months; the point (1, ) represents the number of months, 1, for which the amount to be repaid is $ Eplain 1 Creating Linear Equations in Slope-Intercept Form You can use the slope formula to derive the slope-intercept form of a linear equation Consider a line with slope m and -intercept b The slope formula is m= _ Substitute (, b) for ( 1, 1) and (, ) for (, ) m= -b _ - m = _ -b m = -b m+b= =m+b Multipl both sides b ( ) Add b to both sides Slope-Intercept Form of an Equation If a line has slope m and -intercept (, b), then the line is described b the equation =m+b Module 6 8 Lesson 1 PROFESSIONAL DEVELOPMENT Learning Progressions In this lesson, students build on their understanding of linear functions The focus on the relationships between linear equations and their graphs, including: The slope-intercept form of a linear equation is = m + b, where m represents the slope, and b represents the -intercept A linear function can be graphed b plotting the -intercept and using the slope to find other points that lie on the line The slope-intercept form of a linear equation can be used to write functions that model real-world situations In future lessons, students compare functions represented in different forms Houghton Mifflin Harcourt Publishing Compan EXPLORE Graphing Lines Given Slope and -Intercept INTEGRATE TECHNOLOGY Students have the option of completing the activit either in the book or online CONNECT VOCABULARY Remind students that the word intercept means to come together When a plaer intercepts a football, the plaer and football come together at a certain point Help students make the connection to the -intercept on a graph, the place where the line comes together with the -ais EXPLAIN 1 Creating Linear Equations in Slope-Intercept Form AVOID COMMON ERRORS Some students ma not understand how to use the coordinates ( 1, 1 ) and (, ) to calculate the slope Eplain that the subscripts show which -value goes with which -value; for eample the -value of the first point is 1, the -value of the second point is Remind students that the change in the -coordinates goes in the numerator and the change in -coordinates goes in the denominator Slope-Intercept Form 8

3 Eample 1 Write the equation of each line in slope-intercept form Slope is 3, and (, 5) is on the line Step 1: Find the -intercept = m + b Write the slope intercept form 5 = 3 () + b Substitute 3 for m, for, and 5 for 5 = 6 + b Multipl 5-6 = 6 + b - 6 Subtract 6 from both sides -1 = b Simplif Step : Write the equation = m + b Write the slope intercept form = 3 + (-1) Substitute 3 for m and -1 for b = 3-1 The line passes through (, 5) and (, 13) Step 1: Use the points to find the slope The slope formula is m = _ Substitute (, 5) for ( 1, 1 ) and (, ) for (, ) Houghton Mifflin Harcourt Publishing Compan m = _ 13-5 = _ 8 = - Step : Substitute the slope and - and -coordinates of either of the points in the equation = m + b Substitute for m and the - and -coordinates of the point (, 13) for and 5 Step 3: Substitute for m and for b in the equation = m + b The equation of the line is = = ( ) + b 13 = 8 + b 13-8 = 8 + b = b Module 6 9 Lesson 1 COLLABORATIVE LEARNING Peer-to-Peer Activit Group students in pairs Have each student write slope-intercept equations for four lines: one whose slope is a positive integer, one whose slope is a negative integer, and one whose slope is a fraction Then have partners trade equations Partners should first check that the three conditions are met, then graph the lines 9 Lesson 61

4 Reflect 3 Discussion How would the equation change if (, 5) were used for (, ) and (, 13) were used for ( 1, 1) in the slope formula? Eplain our reasoning The equation would not change at all It doesn t make a difference which point is used for ( 1, 1) and which point is used for (, ) because = Your Turn Write the equation of each line in slope-intercept form Slope is 1, and (3, ) is on the line 5 The line passes through (1, ) and (3, 18) = m + b m = = 1_ = 7 = -1(3) + b = m + b 5 = b = 7 (1) + b The equation of the line is = = b Eplain Graphing from Slope-Intercept Form Writing an equation in slope-intercept form can often make it easier to graph the equation Eample = 5 - Write each equation in slope-intercept form Then graph the line described b the equation The equation = 5 - is alread in slope-intercept form Slope: m = 5 = 5_ 1 -intercept: b = Step 1: Plot (, ) Step : Count 5 units up and 1 unit to the right and plot another point Step 3: Draw a line through the points + 6 = 6 Step 1: Write the equation in slope-intercept form b solving for = 6 - Slope: - 1_ 3 = + Step : Graph the line Plot (, ) Move 1 unit down and 3 units to the right to plot a second point Draw a line through the points - 1_ 3 6 = + 6 -intercept: The equation of the line is = 7-3 Module 6 5 Lesson 1 DIFFERENTIATE INSTRUCTION Communicating Math Have students list the steps for writing a linear function from two given points Sample steps are shown 1 Use the slope formula to find the slope m Substitute m and the coordinates of one point into f () = m + b 3 Solve for the -intercept b Substitute m and b into f () = m + b Houghton Mifflin Harcourt rt Publishing Compan EXPLAIN Graphing from Slope-Intercept Form INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning Eplain to students that one or both intercepts are often used to calculate the slope of a linear equation because the are eas to determine However, an two points that satisf the given equation can be used to determine the slope QUESTIONING STRATEGIES How does the value of b indicate whether the graph is above or below the origin where it intersects the -ais? If b is positive, the -intercept is positive and the graph intersects the -ais above the origin If b is negative, the -intercept is negative and the graph intersects the -ais below the origin What is the advantage of graphing from slope-intercept form? The intercept is one point on the line and a second point can be found easil b using the slope INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections Remind students that slope is the ratio of rise over run Graph a line such as = -_ 1 + in two was, once using a slope of _ -1 and once using a slope of 1_, to show that both result in the same line Slope-Intercept Form 5

5 EXPLAIN 3 Determining Solutions of Equations in Two Variables QUESTIONING STRATEGY For a real-world problem described b a graph of a linear function in which the value of indicates the solution for a given value of, what do ou need to do to solve the problem? Appl the units from the graph to the solution For eample if is time in hours and is cost in dollars, then the solution is dollars for a time of hours INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling Remind students that when time is one of the quantities in a real-world problem, it is usuall the independent variable AVOID COMMON ERRORS Some students ma think that the coefficient of is the slope of the line of the equation regardless of the form of the equation Remind them that if the equation is not in the form = m + b, the coefficient of ma not be the slope Houghton Mifflin Harcourt Publishing Compan Your Turn Write each equation in slope-intercept form Then graph the line described b the equation 6 + = = = 6 = - _ 3 + Eplain 3 Determining Solutions of Equations in Two Variables Given a real-world linear situation described b a table, a graph, or a verbal description, ou can write an equation in slope-intercept form You can use that equation to solve problems Eample 3 Identif the slope and -intercept of the graph that represents each linear situation and interpret what the mean Then write an equation in slope-intercept form and use it to solve the problem For one tai compan, the cost in dollars of a tai ride is a linear function of the distance in miles traveled The initial charge is $5, and the charge per mile is $35 Find the cost of riding a distance of 1 miles The rate of change is $35 per mile, so the slope, m, is 35 The initial cost is the cost to travel miles, $5, so the -intercept, b, is 5 Then an equation is = To find the cost of riding a distance of 1 miles, substitute 1 for and simplif = = 35 (1) + 5 = = 6 (6, 1) is a solution of the equation, and the cost of riding a distance of 1 miles is $6 Module 6 51 Lesson 1 LANGUAGE SUPPORT Connect Vocabular Caution students that a figure called a graph of a line should not be confused with a line graph A line graph is a graph that uses line segments to connect data points A graph of a line is a graph of a linear equation 51 Lesson 61

6 A chairlift descends from a mountain top to pick up skiers at the bottom The height in feet of the chairlift is a linear function of the time in minutes since it begins descending as shown in the graph Find the height of the chairlift minutes after it begins descending Height (ft) Height of a Chairlift (, ) (, 39) (, ) Time (min) The graph contains the points (, ) and (, ) - The slope is = It represents the rate at which the chairlift descends The graph passes through the point (, ), so the -intercept is It represents the height of the chairlift minutes after it begins descending Let be the time in minutes after the chairlift begins to descend Let be the height of the chairlift in feet The equation is = To find the height after minutes, substitute for and simplif = ( ) = -1 + = 39 (, 39) is a solution of the equation, and the height of the chairlift minutes after it begins descending is 39 feet Reflect 8 In the eample involving the tai, how would the equation change if the cost per mile increased or decreased? How would this affect the graph? Increasing the cost per mile would increase the value of m and make the graph steeper Decreasing the cost per mile would decrease the value of m and make the graph less steep Module 6 5 Lesson 1 Slope-Intercept Form 5

7 ELABORATE QUESTIONING STRATEGIES How would ou graph the equation c = 35t + 5? The equation is in slope-intercept form 35 is the slope and 5 is the -intercept Plot the point that corresponds to the -intercept (, 5) Then use the slope to locate a second point on the line Draw a line through the two points SUMMARIZE THE LESSON How do ou write an equation of a line in slope-intercept form when given the slope and -intercept or when given the slope and a point on the line? Using the form = m + b, substitute slope for m and the -intercept for b If ou are given the slope and a point on the line, substitute the slope into = m + b, substitute the coordinates of the point for and, and solve for b Identif the slope and -intercept of the graph that represents each linear situation and interpret what the mean Then write an equation in slope-intercept form and use it to solve the problem Your Turn 9 A local club charges an initial membership fee as well as a monthl cost The cost C in dollars is a linear function of the number of months of membership Find the cost of the membership after months Elaborate Membership Cost Time (months) Cost ($) = = 59 and - 77 = 177 = 59, so the rate of change in the cost is $59 per month The slope, m, is 59 The initial cost is the cost for months, $1, so the -intercept, b, is 1 Let be the number of months and be the cost in dollars The equation is = When =, = 59 () + 1 = 336 So, (, 336) is a solution of the equation, and the membership costs $336 for months 1 What are some advantages to using slope-intercept form? When graphing, it s eas to recognize the slope and -intercept It s also eas to find -values for corresponding -values 11 What are some disadvantages of slope-intercept form? The -intercept ma not be easil visible, and if a -value is given, the -value ma not be easil obtained Houghton Mifflin Harcourt Publishing Compan 1 Essential Question Check-In When given a real-world situation that can be described b a linear equation, how can ou identif the slope and -intercept of the graph of the equation? To find the slope, identif the rate of change for the situation To find the -intercept, identif the initial value for the situation, that is, the value of the dependent variable when the value of the independent value is Module 6 53 Lesson 1 53 Lesson 61