COMMON CORE 5 Locker LESSON Slope-Intercept Form Common Core Math Standards The student is epected to: COMMON CORE F-IF.C.7a Graph linear... functions and show intercepts... Also A-CED.A., A-REI.D. Mathematical Practices COMMON CORE 6. MP.6 Precision Language Objective 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. Houghton Mifflin Harcourt Publishing Compan Image Credits: Echo/ Cultura/Gett Images Name Class Date 6. Slope-Intercept Form Essential Question: How can ou represent a linear function in a wa that reveals its slope and -intercept? 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 -intercept of the graph of the equation that represents the situation is The rate of change in the amount Andrew owes over time is $5 per month. 5 The slope is. Use the -intercept to plot a point on the graph of the equation. The -intercept is, so plot the point (, ). Using the definition of slope, plot a second point. Change in 5 Slope = Change in =_ = 5. Start at the point ou plotted. Count 5 and unit right and plot another point. Draw a line through the points ou plotted. units down Amount ($) Resource Locker Amount Andrew Owes 5 35 3 5 5 5 3 5 6 7 9 Time (Months) Module 6 39 Lesson Name Class Date 6. Slope-Intercept Form Essential Question: How can ou represent a linear function in a wa that reveals its slope and -intercept? 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. Houghton Mifflin Harcourt Publishing Compan Image Credits: Echo/ Cultura/Gett Images F-IF.C.7a Graph linear... functions and show intercepts... Also A-CED.A., A-REI.D. When =, =. The -intercept of the graph of the equation that represents the situation is The rate of change in the amount Andrew owes over time is per month. The slope is. Use the -intercept to plot a point on the graph of the equation. The -intercept is, so plot the point. Using the definition of slope, plot a second point. Change in Slope = Change in =_ =. Start at the point ou plotted. Count units down and unit right and plot another point. Draw a line through the points ou plotted. $5 5 (, ) $ 5 5 5 Amount ($) Resource Amount Andrew Owes 5 35 3 5 5 3 5 6 7 9 Time (Months) Module 6 39 Lesson HARDCOVER PAGES 95 Turn to these pages to find this lesson in the hardcover student edition. 39 Lesson 6.
Reflect. 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, (, 5), and move 5 units down and. How man months will it take Andrew to pa off his loan? Eplain our answer. months; the point (, ) represents the number of months,, for which the amount Eplain 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 (, ) and (, ) for (, ). - b m = _ - - b m = _ m = - b Multipl both sides b ( ). m + b = Slope-Intercept Form of an Equation = m + b Add b to both sides. If a line has slope m and -intercept (, b), then the line is described b the equation = m + b. Eample unit to the right. Then repeat this process beginning at the new point. to be repaid is $. Write the equation of each line in slope-intercept form. Slope is 3, and (, 5) is on the line. Step : 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. - = b Simplif. Step : Write the equation. = m + b Write the slope intercept form. = 3 + (-) Substitute 3 for m and - for b. = 3 - 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 Creating Linear Equations in Slope-Intercept Form AVOID COMMON ERRORS Some students ma not understand how to use the coordinates (, ) 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, the -value of the second point is. Module 6 Lesson 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. Remind students that the change in the -coordinates goes in the numerator and the change in -coordinates goes in the denominator. Slope-Intercept Form
EXPLAIN Graphing from Slope-Intercept Form INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. 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. The line passes through (, 5) and (, 3). Step : Use the points to find the slope. - m = _ - Substitute (, 5) for (, ) and (, ) for (, ). m = _ = _ = Step : Substitute the slope and - and -coordinates of either of the points in the equation = m + b. Step 3: Substitute for m and 5 for b in the equation = m + b. The equation of the line is = + 5. Your Turn Write the equation of each line in slope-intercept form. 3. Slope is, and (3, ) is on the line.. The line passes through (, ) and (3, ). = - (3) + b; 5 = b Eplain Graphing from Slope-Intercept Form Writing an equation in slope-intercept form can make it easier to graph the equation. Eample 3 The equation of the line is = - + 5. 3-5 - Write each equation in slope-intercept form. Then graph the line. 3 = m + b = ( ) + b 3 - = + b - 5 - m = _ 3 - = _ = 7 = 7 () + b; -3 = b = b 3 = + b The equation of the line is = 7-3. 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 PRACTICES Focus on Math Connections MP. Remind students that slope is the ratio of rise over run. Graph a line such as = - _ + in two was, once using a slope of _ - and once using a slope of _, to show that both result in the same line. - Houghton Mifflin Harcourt Publishing Compan = 5 - The equation = 5 - is alread in slope-intercept form. Slope: m = 5 = 5_ -intercept: b = - Step : Plot (, -) Step : Count 5 units up and unit to the right and plot another point. Step 3: Draw a line through the points. - - - - Module 6 Lesson 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. Lesson 6.
+ 6 = 6 Step : Write the equation in slope-intercept form b solving for. + 6 - = 6 - Slope: 6 = - + 6 -intercept: = 3 + Step : Graph the line. Your Turn Plot (, ). Move unit down and units to the right to plot a second point. Draw a line through the points. Write each equation in slope-intercept form. Then graph the line. 5. + = = - + 6. + 3 = 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 miles. The rate of change is $.35 per mile, so the slope, m, is.35. - _ 3 The initial cost is the cost to travel miles, $.5, so the -intercept, b, is.5. 3 - - - - - - - - Houghton Mifflin Harcourt Publishing Compan 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 PRACTICES Focus on Modeling MP. 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. Then an equation is =.35 +.5. =.35 +.5 To find the cost of riding miles, substitute for. =.35 () +.5 = 6 (6, ) is a solution of the equation, and the cost of riding a distance of miles is $6. Module 6 Lesson DIFFERENTIATE INSTRUCTION Communicating Math Have students list the steps for writing a linear function from two given points. Sample steps are shown.. 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. Slope-Intercept Form
ELABORATE B 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. 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. Height (ft) 36 3 6 Height of a Chairlift (, ) (, 39) (, ) 3 5 6 7 9 Time (min) 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. Houghton Mifflin Harcourt Publishing Compan Image Credits: Moodboard/Corbis The graph contains the points (, ) and (, ). - The slope is = -75. - 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 = -75 +. To find the height after minutes, substitute for and simplif. -75 = ( ) + = - + = 39 (, 39) is a solution of the equation, and the height of the chairlift minutes after it begins descending is 39 feet. Reflect 7. 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 3 Lesson 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. 3 Lesson 6.
Your Turn Identif the slope and -intercept of the graph that represents the linear situation and interpret what the mean. Then write an equation in slope-intercept form and use it to solve the problem. EVALUATE. A local club charges an initial membership fee as well as a monthl cost. The cost C in dollars is a linear function of Membership Cost the number of months of membership. Find the cost of the membership after months. Time (months) Cost ($) 77 3 77 m = - 5 = 59; and - 77 = 77 3-6 - 3 3 = 59, 6 5 so the rate of change in the cost is $59 per month. The initial cost is $, so the -intercept, b, is. The equation is = 59 +. f () = 59 () + = 336. So, (, 336) is a solution. Elaborate 9. 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.. 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.. 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. Evaluate: Homework and Practice For each situation, determine the slope and -intercept of the graph of the equation that describes the situation.. John gets a new job and receives a $ signing bonus. After that, he makes $ a da. The rate of change is $ per da, so the slope is. The value of when is (when John has worked das) is $, so the -intercept is. Online Homework Hints and Help Etra Practice Module 6 Lesson Eercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices Houghton Mifflin Harcourt Publishing Compan ASSIGNMENT GUIDE Concepts and Skills Eplore Graphing Lines Given Slope and -Intercept Eample Creating Linear Equations in Slope-Intercept Form Eample Graphing from Slope-Intercept Form Eample 3 Determining Solutions of Equations in Two Variables Practice Eercises Eercises 5 Eercises 5, 6 Eercise 3 5, 7 INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Remind students that the can quickl check that their graphs are reasonable b looking at the slope. Lines with positive slopes rise from left to right, and lines with negative slopes fall from left to right. AVOID COMMON ERRORS Encourage students to use a third point to check a graphed line. The can either choose a point from the graph and check it in the equation, or use the equation to generate a point and check that it is on the graph. Recall of Information MP.6 Precision 5 Skills/Concepts MP.6 Precision 3 Skills/Concepts MP. Modeling 5 6 3 Strategic Thinking MP. Reasoning 7 3 Strategic Thinking MP.3 Logic Slope-Intercept Form
KINESTHETIC EXPERIENCE Use masking tape to outline a coordinate plane on a floor of square tiles. Then give a pair of students a length of rope. Announce an equation in slope-intercept form, and have the two students move around on the plane so that when the hold the rope taut, it represents the line described b the given equation, with each of them as two points on the line. A third student can check that these two points satisf the equation. AVOID COMMON ERRORS Students ma not believe that the have enough information to find the slope of a line. Remind students that, if ou know the equation that describes a line, ou can find its slope b using an two ordered-pair solutions.. Jennifer is miles north of her house, and she is driving north on the highwa at a rate of 55 miles per hour. The rate of change is 55 miles per hour, so the slope is 55. The value of when is (when Jennifer has driven miles) is, so the -intercept is. Sketch a graph that represents the situation. 3. Morwenna rents a truck. She pas $ plus $.5 per mile. Cost ($) 6 Cost of a Rental Truck 6 Distance (mi) Write the equation of each line in slope-intercept form.. An investor invests $ in a certain stock. After the first si months, the value of the stock has increased at a rate of $ per month. 5. Slope is 3, and (, 5) is on the line. 6. Slope is -, and (5, 3) is on the line. 5 = 3 () + b, so = b. The equation is = 3 +. Amount ($) 6 5 5 Value of Investment 3 5 6 7 9 Time (months) 3 = - (5) + b, so 3 = b. The equation is = - + 3. Houghton Mifflin Harcourt Publishing Compan 7. Slope is _, and (, ) is on the line.. Slope is 5, and (, 6) is on the line. = _ () + b, so = b. 6 = 5 () + b, so - = b. The equation is = _ +. The equation is = 5 -. 9. Slope is - _ 3, and (-6, -5) is on the line.. Slope is - _, and (-3, ) is on the line. -5 = - _ (-6) + b, so -9 = b. = - _ 3 The equation is = - _ (-3) _ + b, so = b. 3-9. The equation is = - +. Module 6 5 Lesson 5 Lesson 6.
. Passes through (5, 7) and (3, ). Passes through (-6, ) and (-3, -) m = _ 7 - _ 5-3 = 6 = 3 m = _ - (-) = _ -6 - (-3) -3 = - = 3 (3) + b, so = b. - = -(-3) + b, so - = b. The equation is = 3 -. The equation is = - -. 3. Passes through (6, 6) and (-, ). Passes through (-, -5) and (, 6) 6 - m = 6 - (-) = _ = m = _ -5-6 _ - - = - _ -3 = 3 = _ (-) + b, so 3 = b. 6 = 3 () + b, so - 3 = b. The equation is = _ + 3. The equation is = 3-3. Write each equation in slope-intercept form. Identif the slope and -intercept. Then graph the line described b the equation. INTEGRATE TECHNOLOGY Encourage students to use the function graphing capabilit of a graphing calculator to graph the slope-intercept form and check their answers to the problems. Note that the right side of a function in slope-intercept form can be entered, as the calculator provides Y and the equal sign. Students should eperiment with the function grapher and window settings as well as consult their calculator manuals to learn more. 5. = + 3 = + 3; ; 3 6. = - + - - - - 9. - + = = + 5; ; 5. 3-6 = - - - = - + ; -; - - 7. = _ 3 - = - ; - _ ; - 3 3. = - _ - = - _ - ; - ; - - - = _ + ; ; - - Houghton Mifflin Harcourt Publishing Compan AVOID COMMON ERRORS Students ma have difficult graphing a function that has a fractional rate of change. 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 epressed 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 rightward (the run) to plot a second point given the first point. Module 6 6 Lesson Slope-Intercept Form 6
COLLABORATIVE LEARNING Have students work in groups of three. Give students the following prompt: Connor is on a -da hike. He hiked miles on Da. After hours on Da, he had hiked a total of 6 miles. Have one student draw a graph representing Connor s progress on Da. Have the second student identif the ke features (slope and intercept). Have the third student eplain what the slope and intercept mean in terms of Connor s hike. The -intercept () shows how far he hiked on Da. The slope (.5) shows that he hiked an average of.5 mi/h on Da. JOURNAL Have students show different representations of a linear function: the linear equation, the slope-intercept form of the equation, the graph, and a description of the relationships. Houghton Mifflin Harcourt Publishing Compan Image Credits: YinYang/ Gett Images. -5 - = = - 5_ - ; - 5 ; -. 3 + = - = - 3_ - 3; - 3_ ; -3 - - 3. Sports A figure skating school offers introductor lessons at $5 per session. There is also a registration fee of $3. Write a linear equation in slope-intercept form that represents the situation. You want to take at least 6 lessons. Can ou pa for those lessons using a $ gift certificate? If so, how much mone, if an, will be left on the gift certificate? If not, eplain wh not. The cost per lesson is $5, so the slope of the equation that represents the situation is 5. The initial cost of the lessons (that is, before an lessons are paid for), is $3. So the -intercept is 3. An equation is = 5 + 3. The cost for 6 lessons is 5 (6) + 3 =. A $ gift certificate would pa for the lessons, and $ would be left.. Represent Real World Problems Lorena and Benita are saving mone. The began on the same da. Lorena started with $. Each week she adds $. The graph describes Benita s savings plan. Which girl will have more mone in 6 weeks? How much more will she have? Eplain our reasoning. Lorena; $; the equation for Lorena is = +, so in 6 weeks, she will save $. The equation for Benita is = 5 + 5, so in 6 weeks, she will save $. - - Amount saved ($) 9 7 6 5 3 3 5 6 7 9 Time (weeks) Module 6 7 Lesson 7 Lesson 6.
H.O.T. Focus on Higher Order Thinking 5. Analze Relationships Julio and Jake start their reading assignments the same da. Jake is reading a 6-page book at a rate of pages per da. Julio s book is pages long and his reading rate is times Jake s rate. After 5 das, who will have more pages left to read? How man more? Eplain our reasoning. Jake will have more pages left to read. For Jake, the number of pages to read after das is 6, and the rate of change is -, so the equation is = - + 6. Because -(5) + 6 =, Jake will have pages left to read after 5 das. Julio s pace is 5_ (), so the equation for Julio is = -3 +. Then after 5 das, Julio will have -3(5) + = 3 pages left to read, and - 3 =. So, Jake has - 3 = more pages to read. 6. Eplain the Error John has $ in his bank account when he gets a job. He begins making $7 dollars a da. A student found that the equation that represents this situation is = + 7. What is wrong with the student s equation? Describe and correct the student s error. The student switched the slope and the -intercept. The slope should be 7, and the -intercept should be. So the equation is = 7 +. 7. Justif Reasoning Is it possible to write the equation of ever line in slopeintercept form? Eplain our reasoning. No; it is not possible to write the equation of a vertical line in slope-intercept form. The equation of a vertical line has form = a, where a is a real number. The slope of a vertical line is undefined. Lesson Performance Task The graph shows the cost of a gm membership in each of two ears. What are the values that represent the sign-up fee and the membership monthl fee? How did the values change between the ears? a. Write an equation in slope-intercept form for each of the two lines in the graph. b. What are the values that represent the sign-up fee and the membership cost? How did the values change between the ears? Cost (Dollars) 6 6 Gm Memberships Year Year a. The equation for Year is = +, and the 3 5 6 7 9 equation for Year is = + 6. Time (Months) b. The -intercepts of the two graphs are and 6, so each of these represents the cost when the number of months is, or the sign-up fee. In ear, the -values increase b $ ever month. Since this represents the rate of change in the monthl membership fee, the slope of the line in ear is. In ear, the -values again increase b $ ever month. So the slopes are equal in both ears. This means that the monthl membership fees did not change. However, the sign-up fee increased b $ between ears and. Houghton Mifflin Harcourt Publishing Compan QUESTIONING STRATEGIES How do the graphs of Year and Year compare? The are parallel lines having the same slope, but different -intercepts. How do the -values compare for an whole-number -value? What does this indicate about the costs? is alwas more for Year than Year ; Year costs $ more for an number of months. INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Students can check each ear s equation for correctness b substituting the values of two ordered pairs from the graph of each line into its equation and verifing that both solutions make the equation true. Remind students that checking their equations requires points to define a line, so at least points must be checked. Module 6 Lesson EXTENSION ACTIVITY Have students research the cost of joining two gms. Have students write an equation to represent the cost of each gm. Then have students graph their equations on the same coordinate grid. Students will find that some gms have a higher initial fee, but lower monthl rates than others. Caution students to note whether the fees to attend are weekl or monthl. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. points: Student does not demonstrate understanding of the problem. Slope-Intercept Form